Tuesday, November 25, 2014
For previous remarks on this topic, as it relates to
symmetry axes of the cube, see previous posts tagged Interplay.
The above posts discuss, among other things, the Galois
projective plane of order 3, with 13 points and 13 lines.
These Galois points and lines may be modeled in Euclidean geometry
by the 13 symmetry axes and the 13 rotation planes
of the Euclidean cube. They may also be modeled in Galois geometry
by subsets of the 3x3x3 Galois cube (vector 3-space over GF(3)).
The 3×3×3 Galois Cube
Exercise: Is there any such analogy between the 31 points of the
order-5 Galois projective plane and the 31 symmetry axes of the
Euclidean dodecahedron and icosahedron? Also, how may the
31 projective points be naturally pictured as lines within the
5x5x5 Galois cube (vector 3-space over GF(5))?
Update of Nov. 30, 2014 —
For background to the above exercise, see
pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998), esp.
the citation to a 1983 article by Lemay.
Comments Off on Euclidean-Galois Interplay
Tuesday, July 10, 2012
Comments Off on Euclid vs. Galois
Monday, November 4, 2019
The version of Euclid I.47 in the previous post
suggests a work from a recent Oslo gallery show:
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Wednesday, May 2, 2018
(A sequel to Foster's Space and Sawyer's Space)
See posts now tagged Galois's Space.
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Sunday, November 19, 2017
This is a sequel to yesterday's post Cube Space Continued.
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Saturday, May 20, 2017
Click image to enlarge.
The above 35 projective lines, within a 4×4 array —
The above 15 projective planes, within a 4×4 array (in white) —
* See Galois Tesseract in this journal.
Comments Off on van Lint and Wilson Meet the Galois Tesseract*
Tuesday, May 31, 2016
A very brief introduction:
Comments Off on Galois Space —
Tuesday, January 12, 2016
The above sketch indicates, in a vague, hand-waving, fashion,
a connection between Galois spaces and harmonic analysis.
For more details of the connection, see (for instance) yesterday
afternoon's post Space Oddity.
Comments Off on Harmonic Analysis and Galois Spaces
Tuesday, March 24, 2015
Yesterday's post suggests a review of the following —
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Andries Brouwer, preprint, 1982:
"The Witt designs, Golay codes and Mathieu groups"
(unpublished as of 2013)
Pages 8-9:
Substructures of S(5, 8, 24)
An octad is a block of S(5, 8, 24).
Theorem 5.1
Let B0 be a fixed octad. The 30 octads disjoint from B0
form a self-complementary 3-(16,8,3) design, namely
the design of the points and affine hyperplanes in AG(4, 2),
the 4-dimensional affine space over F2.
Proof….
… (iv) We have AG(4, 2).
(Proof: invoke your favorite characterization of AG(4, 2)
or PG(3, 2), say Dembowski-Wagner or Veblen & Young.
An explicit construction of the vector space is also easy….)
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Related material: Posts tagged Priority.
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Sunday, March 10, 2013
(Continued)
The 16-point affine Galois space:
Further properties of this space:
In Configurations and Squares, see the
discusssion of the Kummer 166 configuration.
Some closely related material:
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Monday, March 4, 2013
Continued from February 27, the day Joseph Frank died…
"Throughout the 1940s, he published essays
and criticism in literary journals, and one,
'Spatial Form in Modern Literature'—
a discussion of experimental treatments
of space and time by Eliot, Joyce, Proust,
Pound and others— published in
The Sewanee Review in 1945, propelled him
to prominence as a theoretician."
— Bruce Weber in this morning's print copy
of The New York Times (p. A15, NY edition)
That essay is reprinted in a 1991 collection
of Frank's work from Rutgers University Press:
See also Galois Space and Occupy Space in this journal.
Frank was best known as a biographer of Dostoevsky.
A very loosely related reference… in a recent Log24 post,
Freeman Dyson's praise of a book on the history of
mathematics and religion in Russia:
"The intellectual drama will attract readers
who are interested in mystical religion
and the foundations of mathematics.
The personal drama will attract readers
who are interested in a human tragedy
with characters who met their fates with
exceptional courage."
Frank is survived by, among others, his wife, a mathematician.
Comments Off on Occupy Galois Space
Thursday, February 21, 2013
(Continued)
The previous post suggests two sayings:
"There is such a thing as a Galois space."
— Adapted from Madeleine L'Engle
"For every kind of vampire, there is a kind of cross."
— Thomas Pynchon
Illustrations—
(Click to enlarge.)
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Sunday, July 29, 2012
(Continued)
The three parts of the figure in today's earlier post "Defining Form"—
— share the same vector-space structure:
| 0 |
c |
d |
c + d |
| a |
a + c |
a + d |
a + c + d |
| b |
b + c |
b + d |
b + c + d |
| a + b |
a + b + c |
a + b + d |
a + b +
c + d |
(This vector-space a b c d diagram is from Chapter 11 of
Sphere Packings, Lattices and Groups , by John Horton
Conway and N. J. A. Sloane, first published by Springer
in 1988.)
The fact that any 4×4 array embodies such a structure was implicit in
the diamond theorem (February 1979). Any 4×4 array, regarded as
a model of the finite geometry AG(4, 2), may be called a Galois tesseract.
(So called because of the Galois geometry involved, and because the
16 cells of a 4×4 array with opposite edges identified have the same
adjacency pattern as the 16 vertices of a tesseract (see, for instance,
Coxeter's 1950 "Self-Dual Configurations and Regular Graphs," figures
5 and 6).)
A 1982 discussion of a more abstract form of AG(4, 2):
Source:
The above 1982 remarks by Brouwer may or may not have influenced
the drawing of the above 1988 Conway-Sloane diagram.
Comments Off on The Galois Tesseract
Thursday, July 12, 2012
An example of lines in a Galois space * —
The 35 lines in the 3-dimensional Galois projective space PG(3,2)—
(Click to enlarge.)
_lines_as_arrays-500w.jpg)
There are 15 different individual linear diagrams in the figure above.
These are the points of the Galois space PG(3,2). Each 3-set of linear diagrams
represents the structure of one of the 35 4×4 arrays and also represents a line
of the projective space.
The symmetry of the linear diagrams accounts for the symmetry of the
840 possible images in the kaleidoscope puzzle.
* For further details on the phrase "Galois space," see
Beniamino Segre's "On Galois Geometries," Proceedings of the
International Congress of Mathematicians, 1958 [Edinburgh].
(Cambridge U. Press, 1960, 488-499.)
(Update of Jan. 5, 2013— This post has been added to finitegeometry.org.)
Comments Off on Galois Space
Saturday, September 3, 2011
A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
two-thirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79T-A37, Notices , Feb. 1979).
Here is some supporting material—
The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.
The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG's
4×4 square as the affine 4-space over the 2-element Galois field.
The passage from Curtis (1974, published in 1976) describes 35 sets
of four "special tetrads" within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4-point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 "special tetrads" rather by the parity
of their intersections with the square's rows and columns.
The affine structure appears in the 1979 abstract mentioned above—
The "35 structures" of the abstract were listed, with an application to
Latin-square orthogonality, in a note from December 1978—
See also a 1987 article by R. T. Curtis—
Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:
“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”
(Received July 20 1987)
– Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353
* For instance:
Update of Sept. 4— This post is now a page at finitegeometry.org.
Comments Off on The Galois Tesseract (continued)
Thursday, September 1, 2011
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Friday, September 17, 2010
Yesterday's excerpt from von Balthasar supplies some Catholic aesthetic background for Galois geometry.
That approach will appeal to few mathematicians, so here is another.
Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace is a book by Leonard Mlodinow published in 2002.
More recently, Mlodinow is the co-author, with Stephen Hawking, of The Grand Design (published on September 7, 2010).
A review of Mlodinow's book on geometry—
"This is a shallow book on deep matters, about which the author knows next to nothing."
— Robert P. Langlands, Notices of the American Mathematical Society, May 2002
The Langlands remark is an apt introduction to Mlodinow's more recent work.
It also applies to Martin Gardner's comments on Galois in 2007 and, posthumously, in 2010.
For the latter, see a Google search done this morning—
Here, for future reference, is a copy of the current Google cache of this journal's "paged=4" page.
Note the link at the bottom of the page in the May 5, 2010, post to Peter J. Cameron's web journal. Following the link, we find…
For n=4, there is only one factorisation, which we can write concisely as 12|34, 13|24, 14|23. Its automorphism group is the symmetric group S4, and acts as S3 on the set of three partitions, as we saw last time; the group of strong automorphisms is the Klein group.
This example generalises, by taking the factorisation to consist of the parallel classes of lines in an affine space over GF(2). The automorphism group is the affine group, and the group of strong automorphisms is its translation subgroup.
See also, in this journal, Window and Window, continued (July 5 and 6, 2010).
Gardner scoffs at the importance of Galois's last letter —
"Galois had written several articles on group theory, and was
merely annotating and correcting those earlier published papers."
— Last Recreations, page 156
For refutations, see the Bulletin of the American Mathematical Society in March 1899 and February 1909.
Comments Off on The Galois Window
Sunday, March 21, 2010
It is well known that the seven (22 + 2 +1) points of the projective plane of order 2 correspond to 2-point subspaces (lines) of the linear 3-space over the two-element field Galois field GF(2), and may be therefore be visualized as 2-cube subsets of the 2×2×2 cube.
Similarly, recent posts* have noted that the thirteen (32 + 3 + 1) points of the projective plane of order 3 may be seen as 3-cube subsets in the 3×3×3 cube.
The twenty-one (42 + 4 +1) points of the (unique) projective plane of order 4 may also be visualized as subsets of a cube– in this case, the 4×4×4 cube. This visualization is somewhat more complicated than the 3×3×3 case, since the 4×4×4 cube has no central subcube, and each projective-plane point corresponds to four, not three, subcubes.
These three cubes, with 8, 27, and 64 subcubes, thus serve as geometric models in a straightforward way– first as models of finite linear spaces, hence as models for small Galois geometries derived from the linear spaces. (The cubes with 8 and 64 subcubes also serve in a less straightforward, and new, way as finite-geometry models– see The Eightfold Cube, Block Designs, and Solomon's Cube.)
A group of collineations** of the 21-point plane is one of two nonisomorphic simple groups of order 20,160. The other is the linear group acting on the linear 4-space over the two-element Galois field GF(2). The 1899 paper establishing the nonisomorphism notes that "the expression Galois Field is perhaps not yet in general use."
Coordinates of the 4×4×4 cube's subcubes can, of course, be regarded as elements of the Galois field GF(64).
The preceding remarks were purely mathematical. The "dreams" of this post's title are not. See…
See also Geometry of the I Ching and a search in this journal for "Galois + Ching."
* February 27 and March 13
** G20160 in Mitchell 1910, LF(3,22) in Edge 1965
— Mitchell, Ulysses Grant, "Geometry and Collineation Groups
of the Finite Projective Plane PG(2,22),"
Princeton Ph.D. dissertation (1910)
— Edge, W. L., "Some Implications of the Geometry of
the 21-Point Plane," Math. Zeitschr. 87, 348-362 (1965)
Comments Off on Galois Field of Dreams
Tuesday, November 5, 2019
A followup to earlier posts on Trudeau vs. Euclid —
Geometry from July 6, 2014:
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Monday, November 4, 2019
Sounds like a story by Optimus Prime —
"Before time began, there was the Cube."
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Previously in Log24: Trudeau and the Story Theory of Truth.
More-recent remarks by Trudeau —
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Bible Stories for Skeptics
Review
With a bit of a twinkle in his eye, Richard Trudeau—a skeptic, retired Unitarian Universalist minister, Harvard Divinity School grad and (though he doesn't use the word) humanist—removes the supernaturalism from some of the common stories in the Judeo-Christian Bible for the edification of non-scholars. He links them to history as known to archeologists and serious historians and tries to salvage some things of value in the collection of diverse materials in the book….
— Edd Doerr, book reviewer for The Unitarian Universalist Humanist Association
About the Author
Richard Trudeau is minister emeritus of the Unitarian Universalist Church of Weymouth, Massachusetts. He holds a Master of Divinity degree with concentration in biblical studies from Harvard Divinity School. He is also professor emeritus of Stonehill College in Easton, Massachusetts, where he specialized in the history of mathematics, the philosophy of science and the history of astronomy. His previous books include Universalism 101 and The Non-Euclidean Revolution.
Product details
File Size: 474 KB
Print Length: 166 pages
Page Numbers Source ISBN: 1493688642
Publisher: Chad Brown Publishing (July 6, 2014)
|
Log24 on the above publication date — July 6, 2014 —
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Comments Off on As Above, So Below*
Tuesday, July 2, 2019
An illustration from the previous post may be interpreted
as an attempt to unbokeh an inscape —
The 15 lines above are Euclidean lines based on pairs within a six-set.
For examples of Galois lines so based, see Six-Set Geometry:
Comments Off on Depth Psychology Meets Inscape Geometry
Sunday, September 9, 2018
"The role of Desargues's theorem was not understood until
the Desargues configuration was discovered. For example,
the fundamental role of Desargues's theorem in the coordinatization
of synthetic projective geometry can only be understood in the light
of the Desargues configuration.
Thus, even as simple a formal statement as Desargues's theorem
is not quite what it purports to be. The statement of Desargues's theorem
pretends to be definitive, but in reality it is only the tip of an iceberg
of connections with other facts of mathematics."
— From p. 192 of "The Phenomenology of Mathematical Proof,"
by Gian-Carlo Rota, in Synthese , Vol. 111, No. 2, Proof and Progress
in Mathematics (May, 1997), pp. 183-196. Published by: Springer.
Stable URL: https://www.jstor.org/stable/20117627.
Related figures —
Note the 3×3 subsquare containing the triangles ABC, etc.
"That in which space itself is contained" — Wallace Stevens
Comments Off on Plan 9 Continues.
Monday, August 27, 2018
From …
Thinking in Four Dimensions
By Dusa McDuff
"I’ve got the rather foolhardy idea of trying to explain
to you the kind of mathematics I do, and the kind of
ideas that seem simple to me. For me, the search
for simplicity is almost synonymous with the search
for structure.
I’m a geometer and topologist, which means that
I study the structure of space …
. . . .
In each dimension there is a simplest space
called Euclidean space … "
— In Roman Kossak, ed.,
Simplicity: Ideals of Practice in Mathematics and the Arts
(Kindle Locations 705-710, 735). Kindle Edition.
For some much simpler spaces of various
dimensions, see Galois Space in this journal.
Comments Off on Geometry and Simplicity
Sunday, April 29, 2018
From the online New York Times this afternoon:
Disney now holds nine of the top 10
domestic openings of all time —
six of which are part of the Marvel
Cinematic Universe. “The result is
a reflection of 10 years of work:
of developing this universe, creating
stakes as big as they were, characters
that matter and stories and worlds that
people have come to love,” Dave Hollis,
Disney’s president of distribution, said
in a phone interview.
From this journal this morning:
"But she felt there must be more to this
than just the sensation of folding space
over on itself. Surely the Centaurs hadn't
spent ten years telling humanity how to
make a fancy amusement-park ride.
There had to be more—"
— Factoring Humanity , by Robert J. Sawyer,
Tom Doherty Associates, 2004 Orb edition,
page 168
"The sensation of folding space . . . ."
Or unfolding:
Click the above unfolded space for some background.
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Monday, March 12, 2018
Remarks related to a recent film and a not-so-recent film.
For some historical background, see Dirac and Geometry in this journal.
Also (as Thas mentions) after Saniga and Planat —
The Saniga-Planat paper was submitted on December 21, 2006.
Excerpts from this journal on that date —
"Open the pod bay doors, HAL."
Comments Off on “Quantum Tesseract Theorem?”
Sunday, March 4, 2018
1955 ("Blackboard Jungle") —
1976 —
2009 —
2016 —
Comments Off on The Square Inch Space: A Brief History
Friday, February 16, 2018
The Institute for Advanced Study (IAS) at Princeton in its Fall 2015 Letter
revived "Beautiful Mathematics" as a title:
This ugly phrase was earlier used by Truman State University
professor Martin Erickson as a book title. See below.
In the same IAS Fall 2015 Letter appear the following remarks
by Freeman Dyson —
". . . a special case of a much deeper connection that Ian Macdonald
discovered between two kinds of symmetry which we call modular and affine.
The two kinds of symmetry were originally found in separate parts of science,
modular in pure mathematics and affine in physics. Modular symmetry is
displayed for everyone to see in the drawings of flying angels and devils
by the artist Maurits Escher. Escher understood the mathematics and got the
details right. Affine symmetry is displayed in the peculiar groupings of particles
created by physicists with high-energy accelerators. The mathematician
Robert Langlands was the first to conjecture a connection between these and
other kinds of symmetry. . . ." (Wikipedia link added.)
The adjective "modular" might aptly be applied to . . .
The adjective "affine" might aptly be applied to . . .
The geometry of the 4×4 square combines modular symmetry
(i.e., related to theta functions) with the affine symmetry above.
Hudson's 1905 discussion of modular symmetry (that of Rosenhain
tetrads and Göpel tetrads) in the 4×4 square used a parametrization
of that square by the digit 0 and the fifteen 2-subsets of a 6-set, but
did not discuss the 4×4 square as an affine space.
For the connection of the 15 Kummer modular 2-subsets with the 16-
element affine space over the two-element Galois field GF(2), see my note
of May 26, 1986, "The 2-subsets of a 6-set are the points of a PG(3,2)" —
— and the affine structure in the 1979 AMS abstract
"Symmetry invariance in a diamond ring" —
For some historical background on the symmetry investigations by
Dyson and Macdonald, see Dyson's 1972 article "MIssed Opportunities."
For Macdonald's own use of the words "modular" and "affine," see
Macdonald, I. G., "Affine Lie algebras and modular forms,"
Séminaire N. Bourbaki , Vol. 23 (1980-1981), Talk no. 577, pp. 258-276.
Comments Off on Two Kinds of Symmetry
Thursday, January 25, 2018
"By an archetype I mean a systematic repertoire
of ideas by means of which a given thinker describes,
by analogical extension , some domain to which
those ideas do not immediately and literally apply."
— Max Black in Models and Metaphors
(Cornell, 1962, p. 241)
"Others … spoke of 'ultimate frames of reference' …."
— Ibid.
A "frame of reference" for the concept four quartets —
A less reputable analogical extension of the same
frame of reference —
Madeleine L'Engle in A Swiftly Tilting Planet :
"… deep in concentration, bent over the model
they were building of a tesseract:
the square squared, and squared again…."
See also the phrase Galois tesseract .
Comments Off on Beware of Analogical Extension
Saturday, September 23, 2017
"With respect to the story's content, the frame thus acts
both as an inclusion of the exterior and as an exclusion
of the interior: it is a perturbation of the outside at the
very core of the story's inside, and as such, it is a blurring
of the very difference between inside and outside."
— Shoshana Felman on a Henry James story, p. 123 in
"Turning the Screw of Interpretation,"
Yale French Studies No. 55/56 (1977), pp. 94-207.
Published by Yale University Press.
See also the previous post and The Galois Tesseract.
Comments Off on The Turn of the Frame
Friday, September 15, 2017
Silas in "Equals" (2015) —
Ever since we were kids it's been drilled into us that …
Our purpose is to explore the universe, you know.
Outer space is where we'll find …
… the answers to why we're here and …
… and where we come from.
Related material —
See also Galois Space in this journal.
Comments Off on Space Art
Sunday, August 27, 2017
The "Black" of the title refers to the previous post.
For the "Well," see Hexagram 48.
Related material —
The Galois Tesseract and, more generally, Binary Coordinate Systems.
Comments Off on Black Well
Saturday, June 3, 2017
Or: The Square
"What we do may be small, but it has
a certain character of permanence."
— G. H. Hardy
* See Expanding the Spielraum in this journal.
Comments Off on Expanding the Spielraum (Continued*)
Tuesday, May 23, 2017
Comments Off on Pursued by a Biplane
Saturday, May 20, 2017
From a review of the 2016 film "Arrival" —
"A seemingly off-hand reference to Abbott and Costello
is our gateway. In a movie as generally humorless as Arrival,
the jokes mean something. Ironically, it is Donnelly, not Banks,
who initiates the joke, naming the verbally inexpressive
Heptapod aliens after the loquacious Classical Hollywood
comedians. The squid-like aliens communicate via those beautiful,
cryptic images. Those signs, when thoroughly comprehended,
open the perceiver to a nonlinear conception of time; this is
Sapir-Whorf taken to the ludicrous extreme."
— Jordan Brower in the Los Angeles Review of Books
Further on in the review —
"Banks doesn’t fully understand the alien language, but she
knows it well enough to get by. This realization emerges
most evidently when Banks enters the alien ship and, floating
alongside Costello, converses with it in their picture-language.
She asks where Abbott is, and it responds — as presented
in subtitling — that Abbott 'is death process.'
'Death process' — dying — is not idiomatic English, and what
we see, written for us, is not a perfect translation but a
rendering of Banks’s understanding. This, it seems to me, is a
crucial moment marking the hard limit of a human mind,
working within the confines of human language to understand
an ultimately intractable xenolinguistic system."
For what may seem like an intractable xenolinguistic system to
those whose experience of mathematics is limited to portrayals
by Hollywood, see the previous post —
van Lint and Wilson Meet the Galois Tesseract.
The death process of van Lint occurred on Sept. 28, 2004.
See this journal on that date.
Comments Off on The Ludicrous Extreme
Tuesday, May 2, 2017
Comments Off on Image Albums
Sunday, February 12, 2017
Euclidean square and triangle—
Galois square and triangle—
For some backstory, see the "preface" of the
previous post and Soifer in this journal.
Comments Off on Religious Art for Sunday
Wednesday, October 5, 2016
From a Google image search yesterday —
Sources (left to right, top to bottom) —
Math Guy (July 16, 2014)
The Galois Tesseract (Sept. 1, 2011)
The Full Force of Roman Law (April 21, 2014)
A Great Moonshine (Sept. 25, 2015)
A Point of Identity (August 8, 2016)
Pascal via Curtis (April 6, 2013)
Correspondences (August 6, 2011)
Symmetric Generation (Sept. 21, 2011)
Comments Off on Sources
Wednesday, August 24, 2016
"That in which space itself is contained" — Wallace Stevens
An image by Steven H. Cullinane from April 1, 2013:
The large Desargues configuration of Euclidean 3-space can be
mapped canonically to the 4×4 square of Galois geometry —
On an Auckland University of Technology thesis by Kate Cullinane —
The thesis reportedly won an Art Directors Club award on April 5, 2013.
Comments Off on Core Statements
Monday, June 27, 2016
From a search in this journal for Euclid + Galois + Interplay —
The 3×3×3 Galois Cube
A tune suggested by the first image above —
Comments Off on Interplay
Saturday, June 18, 2016
In memory of New Yorker artist Anatol Kovarsky,
who reportedly died at 97 on June 1.
Note the Santa, a figure associated with Macy's at Herald Square.
See also posts tagged Herald Square, as well as the following
figure from this journal on the day preceding Kovarsky's death.
A note related both to Galois space and to
the "Herald Square"-tagged posts —
"There is such a thing as a length-16 sequence."
— Saying adapted from a young-adult novel.
Comments Off on Midnight in Herald Square
Sunday, May 8, 2016
Earlier posts have dealt with Solomon Marcus and Solomon Golomb,
both of whom died this year — Marcus on Saint Patrick's Day, and
Golomb on Orthodox Easter Sunday. This suggests a review of
Solomon LeWitt, who died on Catholic Easter Sunday, 2007.
A quote from LeWitt indicates the depth of the word "conceptual"
in his approach to "conceptual art."
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From Sol LeWitt: A Retrospective , edited by Gary Garrels, Yale University Press, 2000, p. 376:
THE SQUARE AND THE CUBE
by Sol LeWitt
"The best that can be said for either the square or the cube is that they are relatively uninteresting in themselves. Being basic representations of two- and three-dimensional form, they lack the expressive force of other more interesting forms and shapes. They are standard and universally recognized, no initiation being required of the viewer; it is immediately evident that a square is a square and a cube a cube. Released from the necessity of being significant in themselves, they can be better used as grammatical devices from which the work may proceed."
"Reprinted from Lucy R. Lippard et al ., “Homage to the Square,” Art in America 55, No. 4 (July-August 1967): 54. (LeWitt’s contribution was originally untitled.)"
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See also the Cullinane models of some small Galois spaces —
Comments Off on The Three Solomons
Friday, May 6, 2016
Thursday, January 21, 2016
My statement yesterday morning that the 15 points
of the finite projective space PG(3,2) are indivisible
was wrong. I was misled by quoting the powerful
rhetoric of Lincoln Barnett (LIFE magazine, 1949).
Points of Euclidean space are of course indivisible:
"A point is that which has no parts" (in some translations).
And the 15 points of PG(3,2) may be pictured as 15
Euclidean points in a square array (with one point removed)
or tetrahedral array (with 11 points added).
The geometry of PG(3,2) becomes more interesting,
however, when the 15 points are each divided into
several parts. For one approach to such a division,
see Mere Geometry. For another approach, click on the
image below.
Comments Off on Dividing the Indivisible
Monday, January 11, 2016
It is an odd fact that the close relationship between some
small Galois spaces and small Boolean spaces has gone
unremarked by mathematicians.
A Google search today for "Galois spaces" + "Boolean spaces"
yielded, apart from merely terminological sources, only some
introductory material I have put on the Web myself.
Some more sophisticated searches, however led to a few
documents from the years 1971 – 1981 …
"Harmonic Analysis of Switching Functions" ,
by Robert J. Lechner, Ch. 5 in A. Mukhopadhyay, editor,
Recent Developments in Switching Theory , Academic Press, 1971.
"Galois Switching Functions and Their Applications,"
by B. Benjauthrit and I. S. Reed,
JPL Deep Space Network Progress Report 42-27 , 1975
D.K. Pradhan, “A Theory of Galois Switching Functions,”
IEEE Trans. Computers , vol. 27, no. 3, pp. 239-249, Mar. 1978
"Switching functions constructed by Galois extension fields,"
by Iwaro Takahashi, Information and Control ,
Volume 48, Issue 2, pp. 95–108, February 1981
An illustration from the Lechner paper above —
"There is such a thing as harmonic analysis of switching functions."
— Saying adapted from a young-adult novel
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Sunday, December 13, 2015
"The colorful story of this undertaking begins with a bang."
— Martin Gardner on the death of Évariste Galois
Comments Off on The Monster as Big as the Ritz
Monday, November 2, 2015
"The office of color in the color line
is a very plain and subordinate one.
It simply advertises the objects of
oppression, insult, and persecution.
It is not the maddening liquor, but
the black letters on the sign
telling the world where it may be had."
— Frederick Douglass, "The Color Line,"
The North American Review , Vol. 132,
No. 295, June 1881, page 575
Or gold letters.
From a search for Seagram in this journal —
"The colorful story of this undertaking begins with a bang."
— Martin Gardner on the death of Évariste Galois
Comments Off on Colorful Story
Wednesday, October 21, 2015
"Perhaps an insane conceit …." Perhaps.
Related remarks on algebra and space —
"The Quality Without a Name" (Log24, August 26, 2015).
Comments Off on Algebra and Space
Wednesday, August 26, 2015
The title phrase, paraphrased without quotes in
the previous post, is from Christopher Alexander's book
The Timeless Way of Building (Oxford University Press, 1979).
A quote from the publisher:
"Now, at last, there is a coherent theory
which describes in modern terms
an architecture as ancient as
human society itself."
Three paragraphs from the book (pp. xiii-xiv):
19. Within this process, every individual act
of building is a process in which space gets
differentiated. It is not a process of addition,
in which preformed parts are combined to
create a whole, but a process of unfolding,
like the evolution of an embryo, in which
the whole precedes the parts, and actualy
gives birth to then, by splitting.
20. The process of unfolding goes step by step,
one pattern at a time. Each step brings just one
pattern to life; and the intensity of the result
depends on the intensity of each one of these
individual steps.
21. From a sequence of these individual patterns,
whole buildings with the character of nature
will form themselves within your thoughts,
as easily as sentences.
Compare to, and contrast with, these illustrations of "Boolean space":
(See also similar illustrations from Berkeley and Purdue.)
Detail of the above image —
Note the "unfolding," as Christopher Alexander would have it.
These "Boolean" spaces of 1, 2, 4, 8, and 16 points
are also Galois spaces. See the diamond theorem —
Comments Off on “The Quality Without a Name”
Saturday, August 15, 2015
Sir Laurence Olivier, in "Term of Trial" (1962), dangles
a participle in front of schoolboy Terence Stamp:
"Walking to school today
my arithmetic book
fell into the gutter"
Were Stamp a Galois, the reply might be "Try this one, sir."
Comments Off on Schoolboy Problem
Friday, August 14, 2015
(A review)
Galois space:
Counting symmetries of Galois space:
The reason for these graphic symmetries in affine Galois space —
symmetries of the underlying projective Galois space:
Comments Off on Discrete Space
Tuesday, June 9, 2015
For geeks* —
" Domain, Domain on the Range , "
where Domain = the Galois tesseract and
Range = the four-element Galois field.
This post was suggested by the previous post,
by a Log24 search for Knight + Move, and by
the phrase "discouraging words" found in that search.
* A term from the 1947 film "Nightmare Alley."
Comments Off on Colorful Song
Thursday, April 16, 2015
"Celebrate National Library Week 2015 (April 12-18, 2015)
with the theme "Unlimited possibilities @ your library®."
See also Library of Hell.
A page from Princeton University Press on March 18, 2012:
… "mathematics and narrative…." (top of page xvii).
I prefer the interplay of Euclidean and Galois mathematics.
Comments Off on National Library Week
Thursday, March 26, 2015
The incidences of points and planes in the
Möbius 84 configuration (8 points and 8 planes,
with 4 points on each plane and 4 planes on each point),
were described by Coxeter in a 1950 paper.*
A table from Monday's post summarizes Coxeter's
remarks, which described the incidences in
spatial terms, with the points and planes as the vertices
and face-planes of two mutually inscribed tetrahedra —
Monday's post, "Gallucci's Möbius Configuration,"
may not be completely intelligible unless one notices
that Coxeter has drawn some of the intersections in his
Fig. 24, a schematic representation of the point-plane
incidences, as dotless, and some as hollow dots. The figure,
"Gallucci's version of Möbius's 84," is shown below.
The hollow dots, representing the 8 points (as opposed
to the 8 planes ) of the configuration, are highlighted in blue.
Here a plane (represented by a dotless intersection) contains
the four points that are represented in the square array as lying
in the same row or same column as the plane.
The above Möbius incidences appear also much earlier in
Coxeter's paper, in figures 6 and 5, where they are shown
as describing the structure of a hypercube.
In figures 6 and 5, the dotless intersections representing
planes have been replaced by solid dots. The hollow dots
have again been highlighted in blue.
Figures 6 and 5 demonstrate the fact that adjacency in the set of
16 vertices of a hypercube is isomorphic to adjacency in the set
of 16 subsquares of a square 4×4 array, provided that opposite
sides of the array are identified, as in Fig. 6. The digits in
Coxeter's labels above may be viewed as naming the positions
of the 1's in (0,1) vectors (x4, x3, x2, x1) over the two-element
Galois field.† In that context, the 4×4 array may be called, instead
of a Möbius hypercube , a Galois tesseract .
* "Self-Dual Configurations and Regular Graphs,"
Bulletin of the American Mathematical Society,
Vol. 56 (1950), pp. 413-455
† The subscripts' usual 1-2-3-4 order is reversed as a reminder
that such a vector may be viewed as labeling a binary number
from 0 through 15, or alternately as labeling a polynomial in
the 16-element Galois field GF(24). See the Log24 post
Vector Addition in a Finite Field (Jan. 5, 2013).
Comments Off on The Möbius Hypercube
Monday, January 5, 2015
Wednesday, March 13, 2013
From a review in the April 2013 issue of
Notices of the American Mathematical Society—
"The author clearly is passionate about mathematics
as an art, as a creative process. In reading this book,
one can easily get the impression that mathematics
instruction should be more like an unfettered journey
into a jungle where an individual can make his or her
own way through that terrain."
From the book under review—
"Every morning you take your machete into the jungle
and explore and make observations, and every day
you fall more in love with the richness and splendor
of the place."
— Lockhart, Paul (2009-04-01).
A Mathematician's Lament:
How School Cheats Us Out of Our Most Fascinating
and Imaginative Art Form (p. 92).
Bellevue Literary Press. Kindle Edition.
Related material: Blackboard Jungle in this journal.
See also Galois Space and Solomon's Mines.
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"I pondered deeply, then, over the
adventures of the jungle. And after
some work with a colored pencil
I succeeded in making my first drawing.
My Drawing Number One.
It looked something like this:
I showed my masterpiece to the
grown-ups, and asked them whether
the drawing frightened them.
But they answered: 'Why should
anyone be frightened by a hat?'"
— The Little Prince
* For the title, see Plato Thanks the Academy (Jan. 3).
Comments Off on Gitterkrieg*
Monday, December 29, 2014
Recent posts tagged Sagan Dodecahedron
mention an association between that Platonic
solid and the 5×5 grid. That grid, when extended
by the six points on a "line at infinity," yields
the 31 points of the finite projective plane of
order five.
For details of how the dodecahedron serves as
a model of this projective plane (PG(2,5)), see
Polster's A Geometrical Picture Book , p. 120:
For associations of the grid with magic rather than
with Plato, see a search for 5×5 in this journal.
Comments Off on Dodecahedron Model of PG(2,5)
Thursday, December 18, 2014
(Five by Five continued)
As the 3×3 grid underlies the order-3 finite projective plane,
whose 13 points may be modeled by
the 13 symmetry axes of the cube,
so the 5×5 grid underlies the order-5 finite projective plane,
whose 31 points may be modeled by
the 31 symmetry axes of the dodecahedron.
See posts tagged Galois-Plane Models.
Comments Off on Platonic Analogy
Wednesday, December 3, 2014
Oslo artist Josefine Lyche has a new Instagram post,
this time on pyramids (the monumental kind).
My response —
Wikipedia's definition of a tetrahedron as a
"triangle-based pyramid" …
… and remarks from a Log24 post of August 14, 2013 :
See also some of Burkard Polster's triangle-based pyramids
and a 1983 triangle-based pyramid in a paper that Polster cites —
(Click image below to enlarge.)
Some other illustrations that are particularly relevant
for Lyche, an enthusiast of magic :
From On Art and Magic (May 5, 2011) —
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Mathematics
The Fano plane block design
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Magic
The Deathly Hallows symbol—
Two blocks short of a design.
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(Updated at about 7 PM ET on Dec. 3.)
Comments Off on Pyramid Dance
Sunday, November 30, 2014
The Regular Tetrahedron
The seven symmetry axes of the regular tetrahedron
are of two types: vertex-to-face and edge-to-edge.
Take these axes as the "points" of a Fano plane.
Each of the tetrahedron's six reflection planes contains
two vertex-to-face axes and one edge-to-edge axis.
Take these six planes as six of the "lines" of a Fano
plane. Then the seventh line is the set of three
edge-to-edge axes.
(The Fano tetrahedron is not original with me.
See Polster's 1998 A Geometrical Picture Book , pp. 16-17.)
The Cube
There are three reflection planes parallel to faces
of the cube. Take the seven nonempty subsets of
the set of these three planes as the "points" of a
Fano plane. Define the Fano "lines" as those triples
of these seven subsets in which each member of
the triple is the symmetric-difference sum of the
other two members.
(This is the eightfold cube discussed at finitegeometry.org.)
Comments Off on Two Physical Models of the Fano Plane
Wednesday, November 26, 2014
Update of Nov. 30, 2014 —
It turns out that the following construction appears on
pages 16-17 of A Geometrical Picture Book , by
Burkard Polster (Springer, 1998).
"Experienced mathematicians know that often the hardest
part of researching a problem is understanding precisely
what that problem says. They often follow Polya's wise
advice: 'If you can't solve a problem, then there is an
easier problem you can't solve: find it.'"
—John H. Conway, foreword to the 2004 Princeton
Science Library edition of How to Solve It , by G. Polya
For a similar but more difficult problem involving the
31-point projective plane, see yesterday's post
"Euclidean-Galois Interplay."
The above new [see update above] Fano-plane model was
suggested by some 1998 remarks of the late Stephen Eberhart.
See this morning's followup to "Euclidean-Galois Interplay"
quoting Eberhart on the topic of how some of the smallest finite
projective planes relate to the symmetries of the five Platonic solids.
Update of Nov. 27, 2014: The seventh "line" of the tetrahedral
Fano model was redefined for greater symmetry.
Comments Off on A Tetrahedral Fano-Plane Model
Update of Nov. 30, 2014 —
For further information on the geometry in
the remarks by Eberhart below, see
pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998). Polster
cites a different article by Lemay.
A search for background to the exercise in the previous post
yields a passage from the late Stephen Eberhart:
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The first three primes p = 2, 3, and 5 therefore yield finite projective planes with 7, 13, and 31 points and lines, respectively. But these are just the numbers of symmetry axes of the five regular solids, as described in Plato's Timaeus : The tetrahedron has 4 pairs of face planes and comer points + 3 pairs of opposite edges, totalling 7 axes; the cube has 3 pairs of faces + 6 pairs of edges + 4 pairs of comers, totalling 13 axes (the octahedron simply interchanges the roles of faces and comers); and the pentagon dodecahedron has 6 pairs of faces + 15 pairs of edges + 10 pairs of comers, totalling 31 axes (the icosahedron again interchanging roles of faces and comers). This is such a suggestive result, one would expect to find it dealt with in most texts on related subjects; instead, while "well known to those who well know such things" (as Richard Guy likes to quip), it is scarcely to be found in the formal literature [9]. The reason for the common numbers, it turns out, is that the groups of symmetry motions of the regular solids are subgroups of the groups of collineations of the respective finite planes, a face axis being different from an edge axis of a regular solid but all points of a projective plane being alike, so the latter has more symmetries than the former.
[9] I am aware only of a series of in-house publications by Fernand Lemay of the Laboratoire de Didactique, Faculté des Sciences de I 'Éducation, Univ. Laval, Québec, in particular those collectively titled Genèse de la géométrie I-X.
— Stephen Eberhart, Dept. of Mathematics,
California State University, Northridge,
"Pythagorean and Platonic Bridges between
Geometry and Algebra," in BRIDGES: Mathematical
Connections in Art, Music, and Science , 1998,
archive.bridgesmathart.org/1998/bridges1998-121.pdf
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Eberhart died of bone cancer in 2003. A memorial by his
high school class includes an Aug. 7, 2003, transcribed
letter from Eberhart to a classmate that ends…
… I earned MA’s in math (UW, Seattle) and history (UM, Missoula) where a math/history PhD program had been announced but canceled. So 1984 to 2002 I taught math (esp. non-Euclidean geometry) at C.S.U. Northridge. It’s been a rich life. I’m grateful.
Steve
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See also another informative BRIDGES paper by Eberhart
on mathematics and the seven traditional liberal arts.
Comments Off on Class Act
Monday, October 27, 2014
A post in honor of Évariste Galois (25 October 1811 – 31 May 1832)
From a book by Richard J. Trudeau titled The Non-Euclidean Revolution —
See also “non-Euclidean” in this journal.
One might argue that Galois geometry, a field ignored by Trudeau,
is also “non-Euclidean,” and (for those who like rhetoric) revolutionary.
Comments Off on Revolutions in Geometry
Saturday, October 25, 2014
In the above illustration of the 3-4-5 Pythagorean triangle,
the grids on each side may be regarded as figures of
Euclidean geometry or of Galois geometry.
In Euclidean geometry, these grids illustrate a property of
the inner triangle.
In elementary Galois geometry, ignoring the connection with
the inner triangle, the grids may be regarded instead as
illustrating vector spaces over finite (i.e., Galois) fields.
Previous posts in this journal have dealt with properties of
the 3×3 and 4×4 grids. This suggests a look at properties of
the next larger grid, the 5×5 array, viewed as a picture of the
two-dimensional vector space (or affine plane) over the finite
Galois field GF(5) (also known as ℤ5).
The 5×5 array may be coordinatized in a natural way, as illustrated
in (for instance) Matters Mathematical , by I.N. Herstein and
Irving Kaplansky, 2nd ed., Chelsea Publishing, 1978, p. 171:
See Herstein and Kaplansky for the elementary Galois geometry of
the 5×5 array.
For 5×5 geometry that is not so elementary, see…
Hafner's abstract:
We describe the Hoffman-Singleton graph geometrically, showing that
it is closely related to the incidence graph of the affine plane over ℤ5.
This allows us to construct all automorphisms of the graph.
The remarks of Brouwer on graphs connect the 5×5-related geometry discussed
by Hafner with the 4×4 geometry related to the Steiner system S(5,8,24).
(See the Miracle Octad Generator of R. T. Curtis and the related coordinatization
by Cullinane of the 4×4 array as a four-dimensional vector space over GF(2).)
Comments Off on Foundation Square
Monday, September 22, 2014
Review of an image from a post of May 6, 2009:
Comments Off on Space
Sunday, September 14, 2014
Structured gray matter:
Graphic symmetries of Galois space:
The reason for these graphic symmetries in affine Galois space —
symmetries of the underlying projective Galois space:
Comments Off on Sensibility
Sunday, August 31, 2014
The Folding
Cynthia Zarin in The New Yorker , issue dated April 12, 2004—
“Time, for L’Engle, is accordion-pleated. She elaborated,
‘When you bring a sheet off the line, you can’t handle it
until it’s folded, and in a sense, I think, the universe can’t
exist until it’s folded — or it’s a story without a book.’”
The geometry of the 4×4 square array is that of the
3-dimensional projective Galois space PG(3,2).
This space occurs, notably, in the Miracle Octad Generator (MOG)
of R. T. Curtis (submitted to Math. Proc. Camb. Phil. Soc. on
15 June 1974). Curtis did not, however, describe its geometric
properties. For these, see the Cullinane diamond theorem.
Some history:
Curtis seems to have obtained the 4×4 space by permuting,
then “folding” 1×8 binary sequences into 4×2 binary arrays.
The original 1×8 sequences came from the method of Turyn
(1967) described by van Lint in his book Coding Theory
(Springer Lecture Notes in Mathematics, No. 201 , first edition
published in 1971). Two 4×2 arrays form each 4×4 square array
within the MOG. This construction did not suggest any discussion
of the geometric properties of the square arrays.
[Rewritten for clarity on Sept. 3, 2014.]
Comments Off on Sunday School
Thursday, July 17, 2014
Continuous Euclidean space to discrete Galois space*
Euclidean space:
From a page by Bryan Clair
Counting symmetries in Euclidean space:
Galois space:
Counting symmetries of Galois space:
The reason for these graphic symmetries in affine Galois space —
symmetries of the underlying projective Galois space:
* For related remarks, see posts of May 26-28, 2012.
Comments Off on Paradigm Shift:
Wednesday, February 5, 2014
Continued from previous post and from Sept. 8, 2009.
Examination of the box's contents does not solve
the contents' real mystery. That requires knowledge
of the non-Euclidean geometry of Galois space.
In this case, without that knowledge, prattle (as in
today's online New York Times ) about creativity and
"thinking outside the box" is pointless.
Comments Off on Mystery Box II
Thursday, November 7, 2013
Yesterday afternoon's post linked to efforts by
the late Robert de Marrais to defend a mathematical
approach to structuralism and kaleidoscopic patterns.
Two examples of non-mathematical discourse on
such patterns:
1. A Royal Society paper from 2012—
Click the above image for related material in this journal.
2. A book by Junichi Toyota from 2009—
Kaleidoscopic Grammar: Investigation into the Nature of Binarism
I find such non-mathematical approaches much less interesting
than those based on the mathematics of reflection groups .
De Marrais described the approaches of Vladimir Arnold and,
earlier, of H. S. M. Coxeter, to such groups. These approaches
dealt only with groups of reflections in Euclidean spaces.
My own interest is in groups of reflections in Galois spaces.
See, for instance, A Simple Reflection Group of Order 168.
Galois spaces over fields of characteristic 2 are particularly
relevant to what Toyota calls binarism .
Comments Off on Pattern Grammar
Monday, April 1, 2013
Background: Rosenhain and Göpel Tetrads in PG(3,2)
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Introduction:
The Large Desargues Configuration
Added by Steven H. Cullinane on Friday, April 19, 2013
Desargues' theorem according to a standard textbook:
"If two triangles are perspective from a point
they are perspective from a line."
The converse, from the same book:
"If two triangles are perspective from a line
they are perspective from a point."
Desargues' theorem according to Wikipedia
combines the above statements:
"Two triangles are in perspective axially [i.e., from a line]
if and only if they are in perspective centrally [i.e., from a point]."
A figure often used to illustrate the theorem,
the Desargues configuration , has 10 points and 10 lines,
with 3 points on each line and 3 lines on each point.
A discussion of the "if and only if" version of the theorem
in light of Galois geometry requires a larger configuration—
15 points and 20 lines, with 3 points on each line
and 4 lines on each point.
This large Desargues configuration involves a third triangle,
needed for the proof (though not the statement ) of the
"if and only if" version of the theorem. Labeled simply
"Desargues' Theorem," the large configuration is the
frontispiece to Volume I (Foundations) of Baker's 6-volume
Principles of Geometry .
Point-line incidence in this larger configuration is,
as noted in the post of April 1 that follows
this introduction, described concisely
by 20 Rosenhain tetrads (defined in 1905 by
R. W. H. T. Hudson in Kummer's Quartic Surface ).
The third triangle, within the larger configuration,
is pictured below.
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A connection discovered today (April 1, 2013)—
(Click to enlarge the image below.)
Update of April 18, 2013
Note that Baker's Desargues-theorem figure has three triangles,
ABC, A'B'C', A"B"C", instead of the two triangles that occur in
the statement of the theorem. The third triangle appears in the
course of proving, not just stating, the theorem (or, more precisely,
its converse). See, for instance, a note on a standard textbook for
further details.
(End of April 18, 2013 update.)
Update of April 14, 2013
See Baker's Proof (Edited for the Web) for a detailed explanation
of the above picture of Baker's Desargues-theorem frontispiece.
(End of April 14, 2013 update.)
Update of April 12, 2013
A different figure, from a site at National Tsing Hua University,
shows the three triangles of Baker's figure more clearly:
(End of update of April 12, 2013)
Update of April 13, 2013
Another in a series of figures illustrating
Desargues's theorem in light of Galois geometry:
See also the original Veblen-Young figure in context.
(End of update of April 13, 2013)
Rota's remarks, while perhaps not completely accurate, provide some context
for the above Desargues-Rosenhain connection. For some other context,
see the interplay in this journal between classical and finite geometry, i.e.
between Euclid and Galois.
For the recent context of the above finite-geometry version of Baker's Vol. I
frontispiece, see Sunday evening's finite-geometry version of Baker's Vol. IV
frontispiece, featuring the Göpel, rather than the Rosenhain, tetrads.
For a 1986 illustration of Göpel and Rosenhain tetrads (though not under
those names), see Picturing the Smallest Projective 3-Space.
In summary… the following classical-geometry figures
are closely related to the Galois geometry PG(3,2):
Volume I of Baker's Principles
has a cover closely related to
the Rosenhain tetrads in PG(3,2)
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Volume IV of Baker's Principles
has a cover closely related to
the Göpel tetrads in PG(3,2)
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Foundations
(click to enlarge)
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Higher Geometry
(click to enlarge)
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Comments Off on Desargues via Rosenhain
Tuesday, February 26, 2013
"I’ve had the privilege recently of being a Harvard University
professor, and there I learned one of the greatest of Harvard
jokes. A group of rabbis are on the road to Golgotha and
Jesus is coming by under the cross. The young rabbi bursts
into tears and says, 'Oh, God, the pity of it!' The old rabbi says,
'What is the pity of it?' The young rabbi says, 'Master, Master,
what a teacher he was.'
'Didn’t publish!'
That cold tenure- joke at Harvard contains a deep truth.
Indeed, Jesus and Socrates did not publish."
— George Steiner, 2002 talk at York University
Related material—
See also Steiner on Galois.
Les Miserables at the Academy Awards
Comments Off on Publication
Tuesday, February 19, 2013
Yesterday's post Permanence dealt with the cube
as a symmetric model of the finite projective plane
PG(2,3), which has 13 points and 13 lines. The points
and lines of the finite geometry occur in the cube as
the 13 axes of symmetry and the 13 planes through
the center perpendicular to those axes. If the three
axes lying in a plane that cuts the cube in a hexagon
are supplemented by the axis perpendicular to that
plane, each plane is associated with four axes and,
dually, each axis is associated with four planes.
My web page on this topic, Cubist Geometries, was
written on February 27, 2010, and first saved to the
Internet Archive on Oct. 4, 2010.
For a more recent treatment of this topic that makes
exactly the same points as the 2010 page, see p. 218
of Configurations from a Graphical Viewpoint , by
Tomaž Pisanski and Brigitte Servatius, published by
Springer on Sept. 23, 2012 (date from both Google
Books and Amazon.com):
For a similar 1998 treatment of the topic, see Burkard Polster's
A Geometrical Picture Book (Springer, 1998), pp. 103-104.
The Pisanski-Servatius book reinforces my argument of Jan. 13, 2013,
that the 13 planes through the cube's center that are perpendicular
to the 13 axes of symmetry of the cube should be called the cube's
symmetry planes , contradicting the usual use of of that term.
That argument concerns the interplay between Euclidean and
Galois geometry. Pisanski and Servatius (and, in 1998, Polster)
emphasize the Euclidean square and cube as guides* to
describing the structure of a Galois space. My Jan. 13 argument
uses Galois structures as a guide to re-describing those of Euclid .
(For a similar strategy at a much more sophisticated level,
see a recent Harvard Math Table.)
Related material: Remarks on configurations in this journal
during the month that saw publication of the Pisanski-Servatius book.
* Earlier guides: the diamond theorem (1978), similar theorems for
2x2x2 (1984) and 4x4x4 cubes (1983), and Visualizing GL(2,p)
(1985). See also Spaces as Hypercubes (2012).
Comments Off on Configurations
Sunday, January 20, 2013
Part I: Synthesis
Part II: Iconic Symbols
Blackboard Jungle , 1955
Part III: Euclid vs. Galois
Comments Off on In the Details
Saturday, January 5, 2013
The finite (i.e., Galois) field GF(16),
according to J. J. Seidel in 1974—
The same field according to Steven H. Cullinane in 1986,
in its guise as the affine 4-space over GF(2)—
The same field, again disguised as an affine 4-space,
according to John H. Conway and N.J.A. Sloane in
Sphere Packings, Lattices, and Groups , first published in 1988—
The above figure by Conway and Sloane summarizes, using
a 4×4 array, the additive vector-space structure of the finite
field GF(16).
This structure embodies what in Euclidean space is called
the parallelogram rule for vector addition—
(Thanks to June Lester for the 3D (uvw) part of the above figure.)
For the transition from this colored Euclidean hypercube
(used above to illustrate the parallelogram rule) to the
4×4 Galois space (illustrated by Cullinane in 1979 and
Conway and Sloane in 1988— or later… I do not have
their book’s first edition), see Diamond Theory in 1937,
Vertex Adjacency in a Tesseract and in a 4×4 Array,
Spaces as Hypercubes, and The Galois Tesseract.
For some related narrative, see tesseract in this journal.
(This post has been added to finitegeometry.org.)
Update of August 9, 2013—
Coordinates for hypercube vertices derived from the
parallelogram rule in four dimensions were better
illustrated by Jürgen Köller in a web page archived in 2002.
Update of August 13, 2013—
The four basis vectors in the 2002 Köller hypercube figure
are also visible at the bottom of the hypercube figure on
page 7 of “Diamond Theory,” excerpts from a 1976 preprint
in Computer Graphics and Art , Vol. 2, No. 1, February 1977.
A predecessor: Coxeter’s 1950 hypercube figure from
“Self-Dual Configurations and Regular Graphs.”
Comments Off on Vector Addition in a Finite Field
Thursday, December 13, 2012
Review:
Cubic models of finite geometries
display an interplay between
Euclidean and Galois geometry.
Related literary remarks: Congregated Light.
Comments Off on Interplay
Monday, August 13, 2012
(An episode of Mathematics and Narrative )
A report on the August 9th opening of Sondheim's Into the Woods—
Amy Adams… explained why she decided to take on the role of the Baker’s Wife.
“It’s the ‘Be careful what you wish’ part,” she said. “Since having a child, I’m really aware that we’re all under a social responsibility to understand the consequences of our actions.” —Amanda Gordon at businessweek.com
Related material—
Amy Adams in Sunshine Cleaning "quickly learns the rules and ropes of her unlikely new market. (For instance, there are products out there specially formulated for cleaning up a 'decomp.')" —David Savage at Cinema Retro
Compare and contrast…
1. The following item from Walpurgisnacht 2012—
2. The six partitions of a tesseract's 16 vertices
into four parallel faces in Diamond Theory in 1937—
Comments Off on Raiders of the Lost Tesseract
Tuesday, May 29, 2012
(Continued from May 29, 2002)
May 29, 1832—
Évariste Galois, Lettre de Galois à M. Auguste Chevalier—
Après cela, il se trouvera, j'espère, des gens qui trouveront leur profit à déchiffrer tout ce gâchis.
(Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.)
Martin Gardner on the above letter—
"Galois had written several articles on group theory, and was merely annotating and correcting those earlier published papers."
– The Last Recreations , by Martin Gardner, published by Springer in 2007, page 156.
Commentary from Dec. 2011 on Gardner's word "published" —
(Click to enlarge.)
Comments Off on The Shining of May 29
Monday, May 28, 2012
Jamie James in The Music of the Spheres
(Springer paperback, 1995), page 28—
Pythagoras constructed a table of opposites
from which he was able to derive every concept
needed for a philosophy of the phenomenal world.
As reconstructed by Aristotle in his Metaphysics,
the table contains ten dualities….
|
Limited
Odd
One
Right
Male
Rest
Straight
Light
Good
Square
|
Unlimited
Even
Many
Left
Female
Motion
Curved
Dark
Bad
Oblong
|
Of these dualities, the first is the most important;
all the others may be seen as different aspects
of this fundamental dichotomy.
For further information, search on peiron + apeiron or
consult, say, Ancient Greek Philosophy , by Vijay Tankha.
The limited-unlimited contrast is not unrelated to the
contrasts between
Comments Off on Fundamental Dichotomy
Monday, May 21, 2012
A web search for the author Cameron McEwen mentioned
in today's noon post was unsuccessful, but it did yield an
essay, quite possibly by a different Cameron McEwen, on
The Digital Wittgenstein:
"The fundamental difference between analog
and digital systems may be understood as
underlying philosophical discourse since the Greeks."
The University of Bergen identifies the Wittgenstein
McEwen as associated with InteLex of Charlottesville.
The title of this post may serve to point out an analogy*
between the InteLex McEwen's analog-digital contrast
and the Euclidean-Galois contrast discussed previously
in this journal.
The latter contrast is exemplified in Pilate Goes to Kindergarten.
* An analogy, as it were, between analogies.
Comments Off on Wittgenstein’s Kindergarten
Thursday, May 3, 2012
(Continued)
“The key is the cocktail that begins the proceedings.”
– Brian Harley, Mate in Two Moves
See also yesterday's Endgame , as well as Play and Interplay
from April 28… and, as a key, the following passage from
an earlier April 28 post—
Euclidean geometry has long been applied
to physics; Galois geometry has not.
The cited webpage describes the interplay
of both sorts of geometry— Euclidean
and Galois, continuous and discrete—
within physical space— if not within
the space of physics . |
Comments Off on Everybody Comes to Rick’s
Saturday, April 28, 2012
A Log24 post, "Bridal Birthday," one year ago today linked to
"The Discrete and the Continuous," a brief essay by David Deutsch.
From that essay—
"The idea of quantization—
the discreteness of physical quantities—
turned out to be immensely fruitful."
Deutsch's "idea of quantization" also appears in
the April 12 Log24 post Mythopoetic—
|
"Is Space Digital?"
— Cover story, Scientific American
magazine, February 2012
"The idea that space may be digital
is a fringe idea of a fringe idea
of a speculative subfield of a subfield."
— Physicist Sabine Hossenfelder
at her weblog on Feb. 5, 2012
"A quantization of space/time
is a holy grail for many theorists…."
— Peter Woit in a comment
at his weblog on April 12, 2012
|
It seems some clarification is in order.
Hossenfelder's "The idea that space may be digital"
and Woit's "a quantization of space/time" may not
refer to the same thing.
Scientific American on the concept of digital space—
"Space may not be smooth and continuous.
Instead it may be digital, composed of tiny bits."
Wikipedia on the concept of quantization—
Causal sets, loop quantum gravity, string theory,
and black hole thermodynamics all predict
a quantized spacetime….
For a purely mathematical approach to the
continuous-vs.-discrete issue, see
Finite Geometry and Physical Space.
The physics there is somewhat tongue-in-cheek,
but the geometry is serious.The issue there is not
continuous-vs.-discrete physics , but rather
Euclidean-vs.-Galois geometry .
Both sorts of geometry are of course valid.
Euclidean geometry has long been applied to
physics; Galois geometry has not. The cited
webpage describes the interplay of both sorts
of geometry— Euclidean and Galois, continuous
and discrete— within physical space— if not
within the space of physics.
Comments Off on Sprechen Sie Deutsch?
Monday, February 20, 2012
In the Beginning…
"As is well known, the Aleph is the first letter of the Hebrew alphabet."
– Borges, "The Aleph" (1945)
From some 1949 remarks of Weyl—
"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."
— Hermann Weyl, "Relativity Theory as a Stimulus in Mathematical Research," Proceedings of the American Philosophical Society , Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949 (Dec. 30, 1949), pp. 535-541
Weyl in 1946—:
"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."
— Hermann Weyl, The Classical Groups , Princeton University Press, 1946, p. 16
Coxeter in 1950 described the elements of the Galois field GF(9) as powers of a primitive root and as ordered pairs of the field of residue-classes modulo 3—
"… the successive powers of the primitive root λ or 10 are
λ = 10, λ2 = 21, λ3 = 22, λ4 = 02,
λ5 = 20, λ6 = 12, λ7 = 11, λ8 = 01.
These are the proper coordinate symbols….
(See Fig. 10, where the points are represented in the Euclidean plane as if the coordinate residue 2 were the ordinary number -1. This representation naturally obscures the collinearity of such points as λ4, λ5, λ7.)"
Coxeter's Figure 10 yields...
The Aleph
The details:
(Click to enlarge)
Coxeter's phrase "in the Euclidean plane" obscures the noncontinuous nature of the transformations that are automorphisms of the above linear 2-space over GF(3).
Comments Off on Coxeter and the Relativity Problem
Saturday, February 18, 2012
From the current Wikipedia article "Symmetry (physics)"—
"In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are 'unchanged', according to a particular observation. A symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is 'preserved' under some change.
A family of particular transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group)."….
"A discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance."
Note the confusion here between continuous (or discontinuous) transformations and "continuous" (or "discontinuous," i.e. "discrete") groups .
This confusion may impede efforts to think clearly about some pure mathematics related to current physics— in particular, about the geometry of spaces made up of individual units ("points") that are not joined together in a continuous manifold.
For an attempt to forestall such confusion, see Noncontinuous Groups.
For related material, see Erlanger and Galois as well as the opening paragraphs of Diamond Theory—
Symmetry is often described as invariance under a group of transformations. An unspoken assumption about symmetry in Euclidean 3-space is that the transformations involved are continuous.
Diamond theory rejects this assumption, and in so doing reveals that Euclidean symmetry may itself be invariant under rather interesting groups of non-continuous (and a-symmetric) transformations. (These might be called noncontinuous groups, as opposed to so-called discontinuous (or discrete ) symmetry groups. See Weyl's Symmetry .)
For example, the affine group A on the 4-space over the 2-element field has a natural noncontinuous and asymmetric but symmetry-preserving action on the elements of a 4×4 array. (Details)
(Version first archived on March 27, 2002)
Update of Sunday, February 19, 2012—
The abuse of language by the anonymous authors
of the above Wikipedia article occurs also in more
reputable sources. For instance—
Some transformations referred to by Brading and Castellani
and their editees as "discrete symmetries" are, in fact, as
linear transformations of continuous spaces, themselves
continuous transformations.
This unfortunate abuse of language is at least made explicit
in a 2003 text, Mathematical Perspectives on Theoretical
Physics (Nirmala Prakash, Imperial College Press)—
"… associated[*] with any given symmetry there always exists
a continuous or a discrete group of transformations….
A symmetry whose associated group is continuous (discrete)
is called a continuous (discrete ) symmetry ." — Pp. 235, 236
[* Associated how?]
Comments Off on Symmetry
Sunday, October 30, 2011
Part I: Timothy Gowers on equivalence relations
Part II: Martin Gardner on normal subgroups
Part III: Evariste Galois on normal subgroups
"In all the history of science there is no completer example
of the triumph of crass stupidity over untamable genius…."
— Eric Temple Bell, Men of Mathematics
See also an interesting definition and Weyl on Galois.
Update of 6:29 PM EDT Oct. 30, 2011—
For further details, see Herstein's phrase
"a tribute to the genius of Galois."
Comments Off on Sermon
Monday, August 29, 2011
A comment today on yesterday's New York Times philosophy column "The Stone"
notes that "Augustine… incorporated Greek ideas of perfection into Christianity."
Yesterday's post here for the Feast of St. Augustine discussed the 2×2×2 cube.
Today's Augustine comment in the Times reflects (through a glass darkly)
a Log24 post from Augustine's Day, 2006, that discusses the larger 4×4×4 cube.
For related material, those who prefer narrative to philosophy may consult
Charles Williams's 1931 novel Many Dimensions . Those who prefer mathematics
to either may consult an interpretation in which Many = Six.
Click image for some background.
Comments Off on Many = Six.
Sunday, August 28, 2011
Yesterday's midday post, borrowing a phrase from the theology of Marvel Comics,
offered Rubik's mechanical contrivance as a rather absurd "Cosmic Cube."
A simpler candidate for the "Cube" part of that phrase:
The Eightfold Cube
As noted elsewhere, a simple reflection group* of order 168 acts naturally on this structure.
"Because of their truly fundamental role in mathematics,
even the simplest diagrams concerning finite reflection groups
(or finite mirror systems, or root systems—
the languages are equivalent) have interpretations
of cosmological proportions."
— Alexandre V. Borovik in "Coxeter Theory: The Cognitive Aspects"
Borovik has a such a diagram—
The planes in Borovik's figure are those separating the parts of the eightfold cube above.
In Coxeter theory, these are Euclidean hyperplanes. In the eightfold cube, they represent three of seven projective points that are permuted by the above group of order 168.
In light of Borovik's remarks, the eightfold cube might serve to illustrate the "Cosmic" part of the Marvel Comics phrase.
For some related theological remarks, see Cube Trinity in this journal.
Happy St. Augustine's Day.
* I.e., one generated by reflections : group actions that fix a hyperplane pointwise. In the eightfold cube, viewed as a vector space of 3 dimensions over the 2-element Galois field, these hyperplanes are certain sets of four subcubes.
Comments Off on The Cosmic Part
Sunday, June 26, 2011
Continued from March 10, 2011 — A post that says
"If Galois geometry is thought of as a paradigm shift
from Euclidean geometry, both… the Kuhn cover
and the nine-point affine plane may be viewed…
as illustrating the shift."
Yesterday's posts The Fano Entity and Theology for Antichristmas,
together with this morning's New York Times obituaries (below)—
—suggest a Sunday School review from last year's
Devil's Night (October 30-31, 2010)—
|
Sunday, October 31, 2010
ART WARS –
m759 @ 2:00 AM
… There is a Cave
Within the Mount of God, fast by his Throne,
Where light and darkness in perpetual round
Lodge and dislodge by turns, which makes through Heav'n
Grateful vicissitude, like Day and Night….
– Paradise Lost , by John Milton
|
See also Ash Wednesday Surprise and Geometry for Jews.
Comments Off on Paradigms Lost
Friday, April 22, 2011
For the title, see Palm Sunday.
"There is a pleasantly discursive treatment of
Pontius Pilate's unanswered question 'What is truth?'" — H. S. M. Coxeter, 1987
From this date (April 22) last year—
|
Richard J. Trudeau in The Non-Euclidean Revolution , chapter on "Geometry and the Diamond Theory of Truth"–
"… Plato and Kant, and most of the philosophers and scientists in the 2200-year interval between them, did share the following general presumptions:
(1) Diamonds– informative, certain truths about the world– exist.
(2) The theorems of Euclidean geometry are diamonds.
Presumption (1) is what I referred to earlier as the 'Diamond Theory' of truth. It is far, far older than deductive geometry."
Trudeau's book was published in 1987. The non-Euclidean* figures above illustrate concepts from a 1976 monograph, also called "Diamond Theory."
Although non-Euclidean,* the theorems of the 1976 "Diamond Theory" are also, in Trudeau's terminology, diamonds.
* "Non-Euclidean" here means merely "other than Euclidean." No violation of Euclid's parallel postulate is implied. |
Trudeau comes to reject what he calls the "Diamond Theory" of truth. The trouble with his argument is the phrase "about the world."
Geometry, a part of pure mathematics, is not about the world. See G. H. Hardy, A Mathematician's Apology .
Comments Off on Romancing the Hyperspace
Sunday, April 17, 2011
The following has rather mysteriously appeared in a search at Google Scholar for "Steven H. Cullinane."
This turns out to be a link to a search within this weblog. I do not know why Google Scholar attributes the resulting web page to a journal article by "AB Story" or why it drew the title from a post within the search and applied it to the entire list of posts found. I am, however, happy with the result— a Palm Sunday surprise with an eclectic mixture of styles that might please the late Robert de Marrais.
I hope the late George Temple would also be pleased. He appears in "Romancing" as a resident of Quarr Abbey, a Benedictine monastery.
The remarks by Martin Hyland quoted in connection with Temple's work are of particular interest in light of the Pope's Christmas remark on mathematics quoted here yesterday.
Comments Off on Annals of Search
Thursday, March 10, 2011
(Continued from February 19)
The cover of the April 1, 1970 second edition of The Structure of Scientific Revolutions , by Thomas S. Kuhn—
This journal on January 19, 2011—
If Galois geometry is thought of as a paradigm shift from Euclidean geometry,
both images above— the Kuhn cover and the nine-point affine plane—
may be viewed, taken together, as illustrating the shift. The nine subcubes
of the Euclidean 3x3x3 cube on the Kuhn cover do not form an affine plane
in the coordinate system of the Galois cube in the second image, but they
at least suggest such a plane. Similarly, transformations of a
non-mathematical object, the 1974 Rubik cube, are not Galois transformations,
but they at least suggest such transformations.
See also today's online Harvard Crimson illustration of problems of translation—
not unrelated to the problems of commensurability discussed by Kuhn.
Comments Off on Paradigms Lost
Tuesday, December 14, 2010
Julie Taymor in an interview published Dec. 12 —
“I’ve got two Broadway shows, a feature film, and Mozart,’’ she said. “It’s a very interesting place to be and to be able to move back and forth, but at a certain point you have to be able to step outside and see,’’ and here she dropped her voice to a tranquil whisper, “it’s just theater. It’s all theater. It’s all theater. The whole thing is theater.’’
Non-theater —
"The interplay between Euclidean and Galois geometry" and
related remarks on interplay — Keats's Laws of Aesthetics.
Part theater, part non-theater —
Cubist crucifixion.
Comments Off on Play and Interplay
Monday, December 13, 2010
Apollo's 13: A Group Theory Narrative —
I. At Wikipedia —
II. Here —
See Cube Spaces and Cubist Geometries.
The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
subcubes in the 27-part (Galois) 3×3×3 cube–
A note from 1985 describing group actions on a 3×3 plane array—
Undated software by Ed Pegg Jr. displays
group actions on a 3×3×3 cube that extend the
3×3 group actions from 1985 described above—
Pegg gives no reference to the 1985 work on group actions.
Comments Off on Mathematics and Narrative continued…
Wednesday, October 20, 2010
"Why the Celebration?"
"Martin Gardner passed away on May 22, 2010."
Imaginary movie poster from stoneship.org
Context— The Gardner Tribute.
Comments Off on Celebration of Mind
Monday, September 27, 2010
… In the Age of Citation
1. INTRODUCTION TO THE PROBLEM
Social network analysis is focused on the patterning of the social
relationships that link social actors. Typically, network data take the
form of a square-actor by actor-binary adjacency matrix, where
each row and each column in the matrix represents a social actor. A
cell entry is 1 if and only if a pair of actors is linked by some social
relationship of interest (Freeman 1989).
— "Using Galois Lattices to Represent Network Data,"
by Linton C. Freeman and Douglas R. White,
Sociological Methodology, Vol. 23, pp. 127–146 (1993)
From this paper's CiteSeer page—
Citations
Visual Image of the Problem—
From a Google search today:
Related material—
"It is better to light one candle…"
"… the early favorite for best picture at the Oscars" — Roger Moore
Comments Off on The Social Network…
Saturday, August 7, 2010
For aficionados of mathematics and narrative —
Illustration from
"The Galois Quaternion— A Story"
This resembles an attempt by Coxeter in 1950 to represent
a Galois geometry in the Euclidean plane—
The quaternion illustration above shows a more natural way to picture this geometry—
not with dots representing points in the Euclidean plane, but rather with unit squares
representing points in a finite Galois affine plane. The use of unit squares to
represent points in Galois space allows, in at least some cases, the actions
of finite groups to be represented more naturally than in Euclidean space.
See Galois Geometry, Geometry Simplified, and
Finite Geometry of the Square and Cube.
Comments Off on The Matrix Reloaded
Thursday, July 22, 2010
Barnes & Noble has an informative new review today of the recent Galois book Duel at Dawn.
It begins…
"In 1820, the Hungarian noble Farkas Bolyai wrote an impassioned cautionary letter to his son Janos:
'I know this way to the very end. I have traversed this bottomless night, which extinguished all light and joy in my life… It can deprive you of your leisure, your health, your peace of mind, and your entire happiness… I turned back when I saw that no man can reach the bottom of this night. I turned back unconsoled, pitying myself and all mankind. Learn from my example…'
Bolyai wasn't warning his son off gambling, or poetry, or a poorly chosen love affair. He was trying to keep him away from non-Euclidean geometry."
For a less dark view (obtained by simply redefining "non-Euclidean" in a more logical way*) see Non-Euclidean Blocks and Finite Geometry and Physical Space.
* Finite geometry is not Euclidean geometry— and is, therefore, non-Euclidean
in the strictest sense (though not according to popular usage), simply because
Euclidean geometry has infinitely many points, and a finite geometry does not.
(This more logical definition of "non-Euclidean" seems to be shared by
at least one other person.)
And some finite geometries are non-Euclidean in the popular-usage sense,
related to Euclid's parallel postulate.
The seven-point Fano plane has, for instance, been called
"a non-Euclidean geometry" not because it is finite
(though that reason would suffice), but because it has no parallel lines.
(See the finite geometry page at the Centre for the Mathematics
of Symmetry and Computation at the University of Western Australia.)
Comments Off on Pilate Goes to Kindergarten, continued
Sunday, June 27, 2010
27
The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
subcubes in the 27-part (Galois) 3×3×3 cube–
Comments Off on Sunday at the Apollo
Thursday, June 24, 2010
Geometry Simplified
(a projective space)
The above finite projective space
is the simplest nontrivial example
of a Galois geometry (i.e., a finite
geometry with coordinates in a
finite (that is, Galois) field.)
The vertical (Euclidean) line represents a
(Galois) point, as does the horizontal line
and also the vertical-and-horizontal
cross that represents the first two points'
binary sum (i.e., symmetric difference,
if the lines are regarded as sets).
Homogeneous coordinates for the
points of this line —
(1,0), (0,1), (1,1).
Here 0 and 1 stand for the elements
of the two-element Galois field GF(2).
The 3-point line is the projective space
corresponding to the affine space
(a plane, not a line) with four points —
(an affine space)
The (Galois) points of this affine plane are
not the single and combined (Euclidean)
line segments that play the role of
points in the 3-point projective line,
but rather the four subsquares
that the line segments separate.
For further details, see Galois Geometry.
There are, of course, also the trivial
two-point affine space and the corresponding
trivial one-point projective space —
Here again, the points of the affine space are
represented by squares, and the point of the
projective space is represented by a line segment
separating the affine-space squares.
Comments Off on Midsummer Noon
Tuesday, June 22, 2010
"By groping toward the light we are made to realize
how deep the darkness is around us."
— Arthur Koestler, The Call Girls: A Tragi-Comedy,
Random House, 1973, page 118
A 1973 review of Koestler's book—
"Koestler's 'call girls,' summoned here and there
by this university and that foundation
to perform their expert tricks, are the butts
of some chilling satire."
Examples of Light—
Felix Christian Klein (1849- June 22, 1925) and Évariste Galois (1811-1832)
Klein on Galois—
"… in France just about 1830 a new star of undreamt-of brilliance— or rather a meteor, soon to be extinguished— lighted the sky of pure mathematics: Évariste Galois."
— Felix Klein, Development of Mathematics in the 19th Century, translated by Michael Ackerman. Brookline, Mass., Math Sci Press, 1979. Page 80.
"… um 1830 herum in Frankreich als ein neuer Stern von ungeahntem Glanze am Himmel der reinen Mathematik aufleuchtet, um freilich, einem Meteor gleich, sehr bald zu verlöschen: Évariste Galois."
— Felix Klein, Vorlesungen Über Die Entwicklung Der Mathematick Im 19. Jahrhundert. New York, Chelsea Publishing Co., 1967. (Vol. I, originally published in Berlin in 1926.) Page 88.
Examples of Darkness—
Martin Gardner on Galois—
"Galois was a thoroughly obnoxious nerd,
suffering from what today would be called
a 'personality disorder.' His anger was
paranoid and unremitting."
Gardner was reviewing a recent book about Galois by one Amir Alexander.
Alexander himself has written some reviews relevant to the Koestler book above.
See Alexander on—
The 2005 Mykonos conference on Mathematics and Narrative
A series of workshops at Banff International Research Station for Mathematical Innovation between 2003 and 2006. "The meetings brought together professional mathematicians (and other mathematical scientists) with authors, poets, artists, playwrights, and film-makers to work together on mathematically-inspired literary works."
Comments Off on Mathematics and Narrative, continued
Monday, June 21, 2010
Cubic models of finite geometries
display an interplay between
Euclidean and Galois geometry.
Example 1— The 2×2×2 Cube—
also known as the eightfold cube—
Group actions on the eightfold cube, 1984—
Version by Laszlo Lovasz et al., 2003—
Lovasz et al. go on to describe the same group actions
as in the 1984 note, without attribution.
Example 2— The 3×3×3 Cube
A note from 1985 describing group actions on a 3×3 plane array—
Undated software by Ed Pegg Jr. displays
group actions on a 3×3×3 cube that extend the
3×3 group actions from 1985 described above—
Pegg gives no reference to the 1985 work on group actions.
Example 3— The 4×4×4 Cube
A note from 27 years ago today—
As far as I know, this version of the
group-actions theorem has not yet been ripped off.
Comments Off on Cube Spaces
Saturday, June 19, 2010
In the above view, four of the tesseract's 16
vertices are overlaid by other vertices.
For views that are more complete and
moveable, see Smith's tesseract page.
Four-Part Tesseract Divisions—
The above figure shows how four-part partitions
of the 16 vertices of a tesseract in an infinite
Euclidean space are related to four-part partitions
of the 16 points in a finite Galois space
Euclidean spaces versus Galois spaces
in a larger context—
Infinite versus Finite
The central aim of Western religion —
"Each of us has something to offer the Creator...
the bridging of
masculine and feminine,
life and death.
It's redemption.... nothing else matters."
-- Martha Cooley in The Archivist (1998)
The central aim of Western philosophy —
Dualities of Pythagoras
as reconstructed by Aristotle:
Limited Unlimited
Odd Even
Male Female
Light Dark
Straight Curved
... and so on ....
"Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy."
— Jamie James in The Music of the Spheres (1993)
|
Another picture related to philosophy and religion—
Jung's Four-Diamond Figure from Aion—
This figure was devised by Jung
to represent the Self. Compare the
remarks of Paul Valéry on the Self—
|
Flight from Eden: The Origins of Modern Literary Criticism and Theory, by Steven Cassedy, U. of California Press, 1990, pages 156-157—
Valéry saw the mind as essentially a relational system whose operation he attempted to describe in the language of group mathematics. "Every act of understanding is based on a group," he says (C, 1:331). "My specialty— reducing everything to the study of a system closed on itself and finite" (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one "group" undergoes a "transformation" and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: "The S[elf] is invariant, origin, locus or field, it's a functional property of consciousness" (C, 15:170 [2:315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind's momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. "Mathematical science… reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind" (O, 1:36). "Psychology is a theory of transformations, we just need to isolate the invariants and the groups" (C, 1:915). "Man is a system that transforms itself" (C, 2:896).
Notes:
O Paul Valéry, Oeuvres (Paris: Pléiade, 1957-60)
C Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 1957-61)
|
Note also the remarks of George David Birkhoff at Rice University
in 1940 (pdf) on Galois's theory of groups and the related
"theory of ambiguity" in Galois's testamentary letter—
|
… metaphysical reasoning always relies on the Principle of Sufficient Reason, and… the true meaning of this Principle is to be found in the “Theory of Ambiguity” and in the associated mathematical “Theory of Groups.”
If I were a Leibnizian mystic, believing in his “preestablished harmony,” and the “best possible world” so satirized by Voltaire in “Candide,” I would say that the metaphysical importance of the Principle of Sufficient Reason and the cognate Theory of Groups arises from the fact that God thinks multi-dimensionally* whereas men can only think in linear syllogistic series, and the Theory of Groups is the appropriate instrument of thought to remedy our deficiency in this respect.
* That is, uses multi-dimensional symbols beyond our grasp.
|
Related material:
Imago Creationis—
A medal designed by Leibniz to show how
binary arithmetic mirrors the creation by God
of something (1) from nothing (0).
Another array of 16 strings of 0's and 1's, this time
regarded as coordinates rather than binary numbers—
Frame of Reference
The Diamond Theorem—
Some context by a British mathematician —
|
Imago
by Wallace Stevens
Who can pick up the weight of Britain,
Who can move the German load
Or say to the French here is France again?
Imago. Imago. Imago.
It is nothing, no great thing, nor man
Of ten brilliancies of battered gold
And fortunate stone. It moves its parade
Of motions in the mind and heart,
A gorgeous fortitude. Medium man
In February hears the imagination's hymns
And sees its images, its motions
And multitudes of motions
And feels the imagination's mercies,
In a season more than sun and south wind,
Something returning from a deeper quarter,
A glacier running through delirium,
Making this heavy rock a place,
Which is not of our lives composed . . .
Lightly and lightly, O my land,
Move lightly through the air again.
|
Comments Off on Imago Creationis
Wednesday, June 2, 2010
"I wonder if there's just been a critical mass
of creepy stories about Harvard
in the last couple of years…
A kind of piling on of
nastiness and creepiness…"
— Margaret Soltan, October 23, 2006
Harvard University Press
on Facebook—
The book that the late Gardner was reviewing
was published in April by Harvard University Press.
If Gardner's remark were true,
Galois would fit right in at Harvard. Example—
The Harvard math department's pie-eating contest—
Comments Off on The Harvard Style
Wikipedia—
"On June 2, Évariste Galois was buried in a common grave of the Montparnasse cemetery whose exact location is unknown."
Évariste Galois, Lettre de Galois à M. Auguste Chevalier—
Après cela, il y aura, j'espère, des gens qui trouveront leur profit à déchiffrer tout ce gâchis.
(Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.)
Martin Gardner on the above letter—
"Galois had written several articles on group theory, and was merely annotating and correcting those earlier published papers."
— The Last Recreations, by Martin Gardner, published by Springer in 2007, page 156.
Leonard E. Dickson—
Comments Off on Rite of Passage
Tuesday, June 1, 2010
"It is a melancholy pleasure that what may be [Martin] Gardner’s last published piece, a review of Amir Alexander’s Duel at Dawn: Heroes, Martyrs & the Rise of Modern Mathematics, will appear next week in our June issue."
– Roger Kimball of The New Criterion, May 23, 2010.
The Gardner piece is now online. It contains…
Gardner's tribute to Galois—
"Galois was a thoroughly obnoxious nerd,
suffering from what today would be called
a 'personality disorder.' His anger was
paranoid and unremitting."
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Comments Off on The Gardner Tribute
Tuesday, May 4, 2010
Romancing the
Non-Euclidean Hyperspace
Backstory —
Mere Geometry, Types of Ambiguity,
Dream Time, and Diamond Theory, 1937
For the 1937 grid, see Diamond Theory, 1937.
The grid is, as Mere Geometry points out, a non-Euclidean hyperspace.
For the diamonds of 2010, see Galois Geometry and Solomon’s Cube.
Comments Off on Mathematics and Narrative, continued
Thursday, April 22, 2010
Stanford Encyclopedia of Philosophy —
“Mereology (from the Greek μερος, ‘part’) is the theory of parthood relations: of the relations of part to whole and the relations of part to part within a whole. Its roots can be traced back to the early days of philosophy, beginning with the Presocratics….”
A non-Euclidean* approach to parts–
Corresponding non-Euclidean*
projective points —
Richard J. Trudeau in The Non-Euclidean Revolution, chapter on “Geometry and the Diamond Theory of Truth”–
“… Plato and Kant, and most of the philosophers and scientists in the 2200-year interval between them, did share the following general presumptions:
(1) Diamonds– informative, certain truths about the world– exist.
(2) The theorems of Euclidean geometry are diamonds.
Presumption (1) is what I referred to earlier as the ‘Diamond Theory’ of truth. It is far, far older than deductive geometry.”
Trudeau’s book was published in 1987. The non-Euclidean* figures above illustrate concepts from a 1976 monograph, also called “Diamond Theory.”
Although non-Euclidean,* the theorems of the 1976 “Diamond Theory” are also, in Trudeau’s terminology, diamonds.
* “Non-Euclidean” here means merely “other than Euclidean.” No violation of Euclid’s parallel postulate is implied.
Comments Off on Mere Geometry
Saturday, March 13, 2010
From yesterday's Seattle Times—
According to police, employees of a Second Avenue mission said the suspect, clad in black and covered in duct tape, had come into the mission "and threatened to blow the place up." He then told staffers "that he was a vampire and wanted to eat people."
The man… also called himself "a space cowboy"….
This suggests two film titles…
Plan 9 from Outer Space—
and Apollo's 13—
The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
subcubes in the (Galois) 3×3×3 cube–
Comments Off on Space Cowboy
Saturday, February 27, 2010
"The cube has…13 axes of symmetry:
6 C2 (axes joining midpoints of opposite edges),
4 C3 (space diagonals), and
3C4 (axes joining opposite face centroids)."
–Wolfram MathWorld article on the cube
These 13 symmetry axes can be used to illustrate the interplay between Euclidean and Galois geometry in a cubic model of the 13-point Galois plane.
The geometer's 3×3×3 cube–
27 separate subcubes unconnected
by any Rubik-like mechanism–
The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
subcubes in the (Galois) 3×3×3 cube–
A closely related structure–
the finite projective plane
with 13 points and 13 lines–
A later version of the 13-point plane
by Ed Pegg Jr.–
A group action on the 3×3×3 cube
as illustrated by a Wolfram program
by Ed Pegg Jr. (undated, but closely
related to a March 26, 1985 note
by Steven H. Cullinane)–
The above images tell a story of sorts.
The moral of the story–
Galois projective geometries can be viewed
in the context of the larger affine geometries
from which they are derived.
The standard definition of points in a Galois projective plane is that they are lines through the (arbitrarily chosen) origin in a corresponding affine 3-space converted to a vector 3-space.
If we choose the origin as the center cube in coordinatizing the 3×3×3 cube (See Weyl's relativity problem ), then the cube's 13 axes of symmetry can, if the other 26 cubes have properly (Weyl's "objectively") chosen coordinates, illustrate nicely the 13 projective points derived from the 27 affine points in the cube model.
The 13 lines of the resulting Galois projective plane may be derived from Euclidean planes through the cube's center point that are perpendicular to the cube's 13 Euclidean symmetry axes.
The above standard definition of points in a Galois projective plane may of course also be used in a simpler structure– the eightfold cube.
(The eightfold cube also allows a less standard way to picture projective points that is related to the symmetries of "diamond" patterns formed by group actions on graphic designs.)
See also Ed Pegg Jr. on finite geometry on May 30, 2006
at the Mathematical Association of America.
Comments Off on Cubist Geometries
Monday, February 1, 2010
"But wait, there's more!"
– Stanley Fish, NY Times Jan. 28
From the editors at The New York Times who, left to their own devices, would produce yet another generation of leftist morons who don't know the difference between education and entertainment–
A new Times column starts today–
The quality of the column's logo speaks for itself. It pictures a cone with dashed lines indicating height and base radius, but unlabeled except for a large italic x to the right of the cone. This enigmatic variable may indicate the cone's height or slant height– or, possibly, its surface area or volume.
Instead of the column's opening load of crap about numbers and Sesame Street, a discussion of its logo might be helpful.
The cone plays a major role in the historical development of mathematics.
Some background from an online edition of Euclid—
"Euclid proved in proposition XII.10 that the cone with the same base and height as a cylinder was one third of the cylinder, but he could not find the ratio of a sphere to the circumscribed cylinder. In the century after Euclid, Archimedes solved this problem as well as the much more difficult problem of the surface area of a sphere."
For Archimedes and the surface area of a sphere, see (for instance) a discussion by Kevin Brown. For more material on Archimedes, see "Archimedes: Volume of a Sphere," by Doug Faires (2001)– Archimedes' heuristic argument from mechanics that involves the volume of a cone– and Archimedes' more rigorous approach in The Works of Archimedes, edited by T. L. Heath (1897).
The work of Euclid and Archimedes on volumes was, of course, long before the discovery of calculus. For a helpful discussion of cone volumes involving high-school-level calculus, see, for instance, the following–
The Times editors apparently feel that
few of their readers are capable of
such high-school-level sophistication.
For some other geometric illustrations
perhaps more appealing than the Times's
dunce cap, see the symbol of
today's saint– a Bridget Cross—
and a web page on
visualized quaternions.
Comments Off on For St. Bridget’s Day
Tuesday, September 8, 2009
Froebel's
Magic Box
Continued from
Dec. 7, 2008,
and from
yesterday.
Non-Euclidean
Blocks
Passages from a classic story:
… he took from his pocket a gadget he had found in the box, and began to unfold it. The result resembled a tesseract, strung with beads….
Tesseract
"Your mind has been conditioned to Euclid," Holloway said. "So this– thing– bores us, and seems pointless. But a child knows nothing of Euclid. A different sort of geometry from ours wouldn't impress him as being illogical. He believes what he sees."
"Are you trying to tell me that this gadget's got a fourth dimensional extension?" Paradine demanded.
"Not visually, anyway," Holloway denied. "All I say is that our minds, conditioned to Euclid, can see nothing in this but an illogical tangle of wires. But a child– especially a baby– might see more. Not at first. It'd be a puzzle, of course. Only a child wouldn't be handicapped by too many preconceived ideas."
"Hardening of the thought-arteries," Jane interjected.
Paradine was not convinced. "Then a baby could work calculus better than Einstein? No, I don't mean that. I can see your point, more or less clearly. Only–"
"Well, look. Let's suppose there are two kinds of geometry– we'll limit it, for the sake of the example. Our kind, Euclidean, and another, which we'll call x. X hasn't much relationship to Euclid. It's based on different theorems. Two and two needn't equal four in it; they could equal y, or they might not even equal. A baby's mind is not yet conditioned, except by certain questionable factors of heredity and environment. Start the infant on Euclid–"
"Poor kid," Jane said.
Holloway shot her a quick glance. "The basis of Euclid. Alphabet blocks. Math, geometry, algebra– they come much later. We're familiar with that development. On the other hand, start the baby with the basic principles of our x logic–"
"Blocks? What kind?"
Holloway looked at the abacus. "It wouldn't make much sense to us. But we've been conditioned to Euclid."
— "Mimsy Were the Borogoves," Lewis Padgett, 1943
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Padgett (pseudonym of a
husband-and-
wife writing team) says that alphabet blocks are the intuitive "basis of
Euclid."
Au contraire; they are the basis of
Gutenberg.
For the intuitive basis of one type of non-Euclidean* geometry– finite geometry over the two-element Galois field– see the work of…
Friedrich Froebel
(1782-1852), who
invented kindergarten.
His "third gift" —
© 2005 The Institute for Figuring
Comments Off on Tuesday September 8, 2009
Wednesday, May 6, 2009
Joke
“My pursuits are a joke
in that the universe is a joke.
One has to reflect
the universe faithfully.”
— John Frederick Michell
Feb. 9, 1933 –
April 24, 2009
“I laugh because I dare not cry.
This is a crazy world and
the only way to enjoy it
is to treat it as a joke.”
— Robert A. Heinlein,
The Number of the Beast
For Marisa Tomei
(born Dec. 4, 1964) —
on the day that
Bob Seger turns 64 —
A Joke:
Points All Her Own
Points All Her Own,
Part I:
(For the backstory, see
the Log24 entries and links
on Marisa Tomei’s birthday
last year.)
Points All Her Own,
Part II:
(For the backstory, see
Galois Geometry:
The Simplest Examples.)
Points All Her Own,
Part III:
(For the backstory, see
Geometry of the I Ching
and the history of
Chinese philosophy.)
In simpler terms:
Smackdown!
Comments Off on Wednesday May 6, 2009
Saturday, April 25, 2009
State of Play
Footprints from California today
(all by a person or persons using Firefox browsers):
7:10 AM
http://m759.xanga.com/679142359/concepts-of-space/?
Concepts of Space: Euclid vs. Galois
8:51 AM
http://m759.xanga.com/689601851/art-wars-continued/?
Art Wars continued: Behind the Picture
1:33 PM
http://m759.xanga.com/678995132/a-riff-for-dave/?
A Riff for Dave: Me and My Shadow
2:11 PM
http://m759.xanga.com/638308002/a-death-of-kings/?
A Death of Kings: In Memory of Bobby Fischer
2:48 PM
http://m759.xanga.com/691644175/art-wars-in-review–/?
Art Wars in review– Through the Looking Glass: A Sort of Eternity
3:28 PM and
http://m759.xanga.com/684680406/annals-of-philosophy/?
Annals of Philosophy: The Dormouse of Perception
4:28 PM
http://m759.xanga.com/641536988/epiphany-for-roy-part-i/?
Epiphany for Roy, Part I
6:03 PM
http://m759.xanga.com/641949564/art-wars-continued/?
At the Still Point: All That Jazz
6:22 PM
http://m759.xanga.com/644330798/where-entertainment-is-not-god/?
Where Entertainment is Not God: The Just Word
7:14 PM
http://m759.xanga.com/643490468/happy-new-yorker-day/?
Happy New Yorker Day– Class Galore
7:16 PM
http://m759.xanga.com/643812753/the-politics-of-change/?
The Politics of Change: Jumpers
"Relax," said the night man.
"We are programmed to receive."
— Hotel California
Comments Off on Saturday April 25, 2009
Thursday, February 5, 2009
Through the
Looking Glass:
A Sort of Eternity
From the new president's inaugural address:
"… in the words of Scripture, the time has come to set aside childish things."
The words of Scripture:
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9
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For we know in part, and we prophesy in part.
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10
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But when that which is perfect is come, then that which is in part shall be done away.
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11
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When I was a child, I spake as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things.
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12
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For now we see through a glass, darkly, but then face to face: now I know in part; but then shall I know even as also I am known.
— First Corinthians 13
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"through a glass"—
[di’ esoptrou].
By means of
a mirror [esoptron].
Childish things:
© 2005 The Institute for Figuring
Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)
Not-so-childish:
Three planes through
the center of a cube
that split it into
eight subcubes:
Through a glass, darkly:
A group of 8 transformations is
generated by affine reflections
in the above three planes.
Shown below is a pattern on
the faces of the 2x2x2 cube
that is symmetric under one of
these 8 transformations–
a 180-degree rotation:
(Click on image
for further details.)
But then face to face:
A larger group of 1344,
rather than 8, transformations
of the 2x2x2 cube
is generated by a different
sort of affine reflections– not
in the infinite Euclidean 3-space
over the field of real numbers,
but rather in the finite Galois
3-space over the 2-element field.
Galois age fifteen,
drawn by a classmate.
These transformations
in the Galois space with
finitely many points
produce a set of 168 patterns
like the one above.
For each such pattern,
at least one nontrivial
transformation in the group of 8
described above is a symmetry
in the Euclidean space with
infinitely many points.
For some generalizations,
see Galois Geometry.
Related material:
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The central aim of Western religion–
"Each of us has something to offer the Creator...
the bridging of
masculine and feminine,
life and death.
It's redemption.... nothing else matters."
-- Martha Cooley in The Archivist (1998)
The central aim of Western philosophy–
Dualities of Pythagoras
as reconstructed by Aristotle:
Limited Unlimited
Odd Even
Male Female
Light Dark
Straight Curved
... and so on ....
"Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy."
— Jamie James in The Music of the Spheres (1993)
"In the garden of Adding
live Even and Odd…
And the song of love's recision
is the music of the spheres."
— The Midrash Jazz Quartet in City of God, by E. L. Doctorow (2000)
A quotation today at art critic Carol Kino's website, slightly expanded:
"Art inherited from the old religion
the power of consecrating things
and endowing them with
a sort of eternity;
museums are our temples,
and the objects displayed in them
are beyond history."
— Octavio Paz,"Seeing and Using: Art and Craftsmanship," in Convergences: Essays on Art and Literature (New York: Harcourt Brace Jovanovich 1987), 52
From Brian O'Doherty's 1976 Artforum essays– not on museums, but rather on gallery space:
"Inside the White Cube"
"We have now reached
a point where we see
not the art but the space first….
An image comes to mind
of a white, ideal space
that, more than any single picture,
may be the archetypal image
of 20th-century art."
"Space: what you
damn well have to see."
— James Joyce, Ulysses
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Comments Off on Thursday February 5, 2009
Wednesday, October 22, 2008
Euclid vs. Galois
On May 4, 2005, I wrote a note about how to visualize the 7-point Fano plane within a cube.
Last month, John Baez showed slides that touched on the same topic. This note is to clear up possible confusion between our two approaches.
From Baez’s Rankin Lectures at the University of Glasgow:
Note that Baez’s statement (
pdf) “Lines in the Fano plane correspond to planes through the origin [the vertex labeled ‘1’] in this cube” is, if taken (wrongly) as a statement about a cube in
Euclidean 3-space, false.
The statement is, however, true of the eightfold cube, whose eight subcubes correspond to points of the linear 3-space over the two-element field, if “planes through the origin” is interpreted as planes within that linear 3-space, as in Galois geometry, rather than within the Euclidean cube that Baez’s slides seem to picture.
This Galois-geometry interpretation is, as an article of his from 2001 shows, actually what Baez was driving at. His remarks, however, both in 2001 and 2008, on the plane-cube relationship are both somewhat trivial– since “planes through the origin” is a standard definition of lines in projective geometry– and also unrelated– apart from the possibility of confusion– to my own efforts in this area. For further details, see The Eightfold Cube.
Comments Off on Wednesday October 22, 2008
Friday, November 24, 2006
Galois’s Window:
Geometry
from Point
to Hyperspace
by Steven H. Cullinane
Euclid is “the most famous
geometer ever known
and for good reason:
for millennia it has been
his window
that people first look through
when they view geometry.”
— Euclid’s Window:
The Story of Geometry
from Parallel Lines
to Hyperspace,
by Leonard Mlodinow
“…the source of
all great mathematics
is the special case,
the concrete example.
It is frequent in mathematics
that every instance of a
concept of seemingly
great generality is
in essence the same as
a small and concrete
special case.”
— Paul Halmos in
I Want To Be a Mathematician
Euclid’s geometry deals with affine
spaces of 1, 2, and 3 dimensions
definable over the field
of real numbers.
Each of these spaces
has infinitely many points.
Some simpler spaces are those
defined over a finite field–
i.e., a “Galois” field–
for instance, the field
which has only two
elements, 0 and 1, with
addition and multiplication
as follows:
We may picture the smallest
affine spaces over this simplest
field by using square or cubic
cells as “points”:
From these five finite spaces,
we may, in accordance with
Halmos’s advice,
select as “a small and
concrete special case”
the 4-point affine plane,
which we may call
Galois’s Window.
The interior lines of the picture
are by no means irrelevant to
the space’s structure, as may be
seen by examining the cases of
the above Galois affine 3-space
and Galois affine hyperplane
in greater detail.
For more on these cases, see
The Eightfold Cube,
Finite Relativity,
The Smallest Projective Space,
Latin-Square Geometry, and
Geometry of the 4×4 Square.
(These documents assume that
the reader is familar with the
distinction between affine and
projective geometry.)
These 8- and 16-point spaces
may be used to
illustrate the action of Klein’s
simple group of order 168
and the action of
a subgroup of 322,560 elements
within the large Mathieu group.
The view from Galois’s window
also includes aspects of
quantum information theory.
For links to some papers
in this area, see
Elements of Finite Geometry.
Comments Off on Friday November 24, 2006
Tuesday, October 31, 2006
"The show has an endgame, end-time mood….
I would call all these strategies fear of form…. the dismissal of originality is perhaps the oldest ploy in the postmodern playbook. To call yourself an artist at all is by definition to announce a faith, however unacknowledged, in some form of originality, first for yourself, second, perhaps, for the rest of us.
Fear of form above all means fear of compression– of an artistic focus that condenses experiences, ideas and feelings into something whole, committed and visually comprehensible."
— Roberta Smith
It is doubtful that Smith
would consider the
following "found" art an
example of originality.
It nevertheless does
"announce a faith."
"First for yourself"
Today's mid-day
Pennsylvania number:
707
See Log24 on 7/07
and the above review.
"Second, perhaps,
for the rest of us"
Today's evening
Pennsylvania number:
384
This number is an
example of what the
reviewer calls "compression"–
"an artistic focus that condenses
experiences, ideas and feelings
into something
whole, committed
and visually comprehensible."
"Experiences"
See (for instance)
Joan Didion's writings
(1160 pages, 2.35 pounds)
on "the shifting phantasmagoria
which is our actual experience."
Comments Off on Tuesday October 31, 2006
Wednesday, March 29, 2006
Note: Carmichael's reference is to
A. Emch, "Triple and multiple systems, their geometric configurations and groups," Trans. Amer. Math. Soc. 31 (1929), 25–42.
"There is such a thing as a tesseract."
— A Wrinkle in Time
Comments Off on Wednesday March 29, 2006
Sunday, November 20, 2005
“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?'”
— H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau’s remarks on the “Story Theory” of truth as opposed to the “Diamond Theory” of truth in The Non-Euclidean Revolution
“A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the ‘Story Theory’ of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called ‘true.’ The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes*….”
— Richard J. Trudeau in
The Non-Euclidean Revolution
“‘Deniers’ of truth… insist that each of us is trapped in his own point of view; we make up stories about the world and, in an exercise of power, try to impose them on others.”
— Jim Holt in The New Yorker.
(Click on the box below.)
Exercise of Power:
Show that a white horse–
a figure not unlike the
symbol of the mathematics
publisher Springer–
is traced, within a naturally
arranged rectangular array of
polynomials, by the powers of x
modulo a polynomial
irreducible over a Galois field.
This horse, or chess knight–
“Springer,” in German–
plays a role in “Diamond Theory”
(a phrase used in finite geometry
in 1976, some years before its use
by Trudeau in the above book).
Related material
On this date:
In 1490, The White Knight
(Tirant lo Blanc 
)–
a major influence on Cervantes–
was published, and in 1910
the Mexican Revolution began.
Illustration:
Zapata by Diego Rivera,
Museum of Modern Art,
New York
Description from Amazon.com—
“First published in the Catalan language in Valencia in 1490…. Reviewing the first modern Spanish translation in 1969 (Franco had ruthlessly suppressed the Catalan language and literature), Mario Vargas Llosa hailed the epic’s author as ‘the first of that lineage of God-supplanters– Fielding, Balzac, Dickens, Flaubert, Tolstoy, Joyce, Faulkner– who try to create in their novels an all-encompassing reality.'”
Comments Off on Sunday November 20, 2005
Saturday, June 4, 2005
Drama of the Diagonal The 4×4 Square: French Perspectives
Earendil_Silmarils:
Les Anamorphoses:
“Pour construire un dessin en perspective,
le peintre trace sur sa toile des repères:
la ligne d’horizon (1),
le point de fuite principal (2)
où se rencontre les lignes de fuite (3)
et le point de fuite des diagonales (4).”
_______________________________
Serge Mehl,
Perspective &
Géométrie Projective:
“… la géométrie projective était souvent
synonyme de géométrie supérieure.
Elle s’opposait à la géométrie
euclidienne: élémentaire…
La géométrie projective, certes supérieure
car assez ardue, permet d’établir
de façon élégante des résultats de
la géométrie élémentaire.”
Similarly…
Finite projective geometry
(in particular, Galois geometry)
is certainly superior to
the elementary geometry of
quilt-pattern symmetry
and allows us to establish
de façon élégante
some results of that
elementary geometry.
Other Related Material…
from algebra rather than
geometry, and from a German
rather than from the French:
“This is the relativity problem:
to fix objectively a class of
equivalent coordinatizations
and to ascertain
the group of transformations S
mediating between them.”
— Hermann Weyl,
The Classical Groups,
Princeton U. Press, 1946
Evariste Galois
Weyl also says that the profound branch
of mathematics known as Galois theory
“… is nothing else but the
relativity theory for the set Sigma,
a set which, by its discrete and
finite character, is conceptually
so much simpler than the
infinite set of points in space
or space-time dealt with
by ordinary relativity theory.”
— Weyl, Symmetry,
Princeton U. Press, 1952
Metaphor and Algebra…
“Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra.”
— attributed, in varying forms, to
Max Black, Models and Metaphors, 1962
For metaphor and
algebra combined, see
“Symmetry invariance in a diamond ring,”
A.M.S. abstract 79T-A37,
Notices of the
American Mathematical Society,
February 1979, pages A-193, 194 —
the original version of the 4×4 case
of the diamond theorem.
“When approaching unfamiliar territory, we often, as observed earlier, try to describe or frame the novel situation using metaphors based on relations perceived in a familiar domain, and by using our powers of association, and our ability to exploit the structural similarity, we go on to conjecture new features for consideration, often not noticed at the outset. The metaphor works, according to Max Black, by transferring the associated ideas and implications of the secondary to the primary system, and by selecting, emphasising and suppressing features of the primary in such a way that new slants on it are illuminated.”
— Paul Thompson, University College, Oxford,
The Nature and Role of Intuition
in Mathematical Epistemology
A New Slant…
That intuition, metaphor (i.e., analogy), and association may lead us astray is well known. The examples of French perspective above show what might happen if someone ignorant of finite geometry were to associate the phrase “4×4 square” with the phrase “projective geometry.” The results are ridiculously inappropriate, but at least the second example does, literally, illuminate “new slants”– i.e., diagonals– within the perspective drawing of the 4×4 square.
Similarly, analogy led the ancient Greeks to believe that the diagonal of a square is commensurate with the side… until someone gave them a new slant on the subject.
Comments Off on Saturday June 4, 2005
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